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Theorem 2ndcctbss 21258
Description: If a topology is second-countable, every base has a countable subset which is a base. Exercise 16B2 in Willard. (Contributed by Jeff Hankins, 28-Jan-2010.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
Hypotheses
Ref Expression
2ndcctbss.1 |- X = U. B
2ndcctbss.2 |- J = ( topGen ` B )
2ndcctbss.3 |- S = { <. u , v >. | ( u e. c /\ v e. c /\ E. w e. B ( u C_ w /\ w C_ v ) ) }
Assertion
Ref Expression
2ndcctbss |- ( ( B e. TopBases /\ J e. 2ndc ) -> E. b e. TopBases ( b ~<_ om /\ b C_ B /\ J = ( topGen ` b ) ) )
Distinct variable groups:   b, c, u, v, w, B   J, b, c
Allowed substitution hints:   S( w, v, u, b, c)   J( w, v, u)   X( w, v, u, b, c)
Proof of Theorem 2ndcctbss
Dummy variables d f m n o t x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 477 . . 3 |- ( ( B e. TopBases /\ J e. 2ndc ) -> J e. 2ndc )
2 is2ndc 21249 . . 3 |- ( J e. 2ndc <-> E. c e. TopBases ( c ~<_ om /\ ( topGen ` c ) = J ) )
31, 2sylib 208 . 2 |- ( ( B e. TopBases /\ J e. 2ndc ) -> E. c e. TopBases ( c ~<_ om /\ ( topGen ` c ) = J ) )
4 vex 3203 . . . . . . 7 |- c e. _V
54, 4xpex 6962 . . . . . 6 |- ( c X. c ) e. _V
6 3simpa 1058 . . . . . . . 8 |- ( ( u e. c /\ v e. c /\ E. w e. B ( u C_ w /\ w C_ v ) ) -> ( u e. c /\ v e. c ) )
76ssopab2i 5003 . . . . . . 7 |- { <. u , v >. | ( u e. c /\ v e. c /\ E. w e. B ( u C_ w /\ w C_ v ) ) } C_ { <. u , v >. | ( u e. c /\ v e. c ) }
8 2ndcctbss.3 . . . . . . 7 |- S = { <. u , v >. | ( u e. c /\ v e. c /\ E. w e. B ( u C_ w /\ w C_ v ) ) }
9 df-xp 5120 . . . . . . 7 |- ( c X. c ) = { <. u , v >. | ( u e. c /\ v e. c ) }
107, 8, 93sstr4i 3644 . . . . . 6 |- S C_ ( c X. c )
11 ssdomg 8001 . . . . . 6 |- ( ( c X. c ) e. _V -> ( S C_ ( c X. c ) -> S ~<_ ( c X. c ) ) )
125, 10, 11mp2 9 . . . . 5 |- S ~<_ ( c X. c )
134xpdom1 8059 . . . . . . . . 9 |- ( c ~<_ om -> ( c X. c ) ~<_ ( om X. c ) )
14 omex 8540 . . . . . . . . . 10 |- om e. _V
1514xpdom2 8055 . . . . . . . . 9 |- ( c ~<_ om -> ( om X. c ) ~<_ ( om X. om ) )
16 domtr 8009 . . . . . . . . 9 |- ( ( ( c X. c ) ~<_ ( om X. c ) /\ ( om X. c ) ~<_ ( om X. om ) ) -> ( c X. c ) ~<_ ( om X. om ) )
1713, 15, 16syl2anc 693 . . . . . . . 8 |- ( c ~<_ om -> ( c X. c ) ~<_ ( om X. om ) )
18 xpomen 8838 . . . . . . . 8 |- ( om X. om ) ~~ om
19 domentr 8015 . . . . . . . 8 |- ( ( ( c X. c ) ~<_ ( om X. om ) /\ ( om X. om ) ~~ om ) -> ( c X. c ) ~<_ om )
2017, 18, 19sylancl 694 . . . . . . 7 |- ( c ~<_ om -> ( c X. c ) ~<_ om )
2120adantr 481 . . . . . 6 |- ( ( c ~<_ om /\ ( topGen ` c ) = J ) -> ( c X. c ) ~<_ om )
2221ad2antll 765 . . . . 5 |- ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) -> ( c X. c ) ~<_ om )
23 domtr 8009 . . . . 5 |- ( ( S ~<_ ( c X. c ) /\ ( c X. c ) ~<_ om ) -> S ~<_ om )
2412, 22, 23sylancr 695 . . . 4 |- ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) -> S ~<_ om )
258relopabi 5245 . . . . . . . . 9 |- Rel S
26 simpr 477 . . . . . . . . 9 |- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ x e. S ) -> x e. S )
27 1st2nd 7214 . . . . . . . . 9 |- ( ( Rel S /\ x e. S ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
2825, 26, 27sylancr 695 . . . . . . . 8 |- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ x e. S ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
2928, 26eqeltrrd 2702 . . . . . . 7 |- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ x e. S ) -> <. ( 1st ` x ) , ( 2nd ` x ) >. e. S )
30 df-br 4654 . . . . . . . . 9 |- ( ( 1st ` x ) S ( 2nd ` x ) <-> <. ( 1st ` x ) , ( 2nd ` x ) >. e. S )
31 fvex 6201 . . . . . . . . . 10 |- ( 1st ` x ) e. _V
32 fvex 6201 . . . . . . . . . 10 |- ( 2nd ` x ) e. _V
33 simpl 473 . . . . . . . . . . . 12 |- ( ( u = ( 1st ` x ) /\ v = ( 2nd ` x ) ) -> u = ( 1st ` x ) )
3433eleq1d 2686 . . . . . . . . . . 11 |- ( ( u = ( 1st ` x ) /\ v = ( 2nd ` x ) ) -> ( u e. c <-> ( 1st ` x ) e. c ) )
35 simpr 477 . . . . . . . . . . . 12 |- ( ( u = ( 1st ` x ) /\ v = ( 2nd ` x ) ) -> v = ( 2nd ` x ) )
3635eleq1d 2686 . . . . . . . . . . 11 |- ( ( u = ( 1st ` x ) /\ v = ( 2nd ` x ) ) -> ( v e. c <-> ( 2nd ` x ) e. c ) )
37 sseq1 3626 . . . . . . . . . . . . 13 |- ( u = ( 1st ` x ) -> ( u C_ w <-> ( 1st ` x ) C_ w ) )
38 sseq2 3627 . . . . . . . . . . . . 13 |- ( v = ( 2nd ` x ) -> ( w C_ v <-> w C_ ( 2nd ` x ) ) )
3937, 38bi2anan9 917 . . . . . . . . . . . 12 |- ( ( u = ( 1st ` x ) /\ v = ( 2nd ` x ) ) -> ( ( u C_ w /\ w C_ v ) <-> ( ( 1st ` x ) C_ w /\ w C_ ( 2nd ` x ) ) ) )
4039rexbidv 3052 . . . . . . . . . . 11 |- ( ( u = ( 1st ` x ) /\ v = ( 2nd ` x ) ) -> ( E. w e. B ( u C_ w /\ w C_ v ) <-> E. w e. B ( ( 1st ` x ) C_ w /\ w C_ ( 2nd ` x ) ) ) )
4134, 36, 403anbi123d 1399 . . . . . . . . . 10 |- ( ( u = ( 1st ` x ) /\ v = ( 2nd ` x ) ) -> ( ( u e. c /\ v e. c /\ E. w e. B ( u C_ w /\ w C_ v ) ) <-> ( ( 1st ` x ) e. c /\ ( 2nd ` x ) e. c /\ E. w e. B ( ( 1st ` x ) C_ w /\ w C_ ( 2nd ` x ) ) ) ) )
4231, 32, 41, 8braba 4992 . . . . . . . . 9 |- ( ( 1st ` x ) S ( 2nd ` x ) <-> ( ( 1st ` x ) e. c /\ ( 2nd ` x ) e. c /\ E. w e. B ( ( 1st ` x ) C_ w /\ w C_ ( 2nd ` x ) ) ) )
4330, 42bitr3i 266 . . . . . . . 8 |- ( <. ( 1st ` x ) , ( 2nd ` x ) >. e. S <-> ( ( 1st ` x ) e. c /\ ( 2nd ` x ) e. c /\ E. w e. B ( ( 1st ` x ) C_ w /\ w C_ ( 2nd ` x ) ) ) )
4443simp3bi 1078 . . . . . . 7 |- ( <. ( 1st ` x ) , ( 2nd ` x ) >. e. S -> E. w e. B ( ( 1st ` x ) C_ w /\ w C_ ( 2nd ` x ) ) )
4529, 44syl 17 . . . . . 6 |- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ x e. S ) -> E. w e. B ( ( 1st ` x ) C_ w /\ w C_ ( 2nd ` x ) ) )
46 fvi 6255 . . . . . . . 8 |- ( B e. TopBases -> ( _I ` B ) = B )
4746ad3antrrr 766 . . . . . . 7 |- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ x e. S ) -> ( _I ` B ) = B )
4847rexeqdv 3145 . . . . . 6 |- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ x e. S ) -> ( E. w e. ( _I ` B ) ( ( 1st ` x ) C_ w /\ w C_ ( 2nd ` x ) ) <-> E. w e. B ( ( 1st ` x ) C_ w /\ w C_ ( 2nd ` x ) ) ) )
4945, 48mpbird 247 . . . . 5 |- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ x e. S ) -> E. w e. ( _I ` B ) ( ( 1st ` x ) C_ w /\ w C_ ( 2nd ` x ) ) )
5049ralrimiva 2966 . . . 4 |- ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) -> A. x e. S E. w e. ( _I ` B ) ( ( 1st ` x ) C_ w /\ w C_ ( 2nd ` x ) ) )
51 fvex 6201 . . . . 5 |- ( _I ` B ) e. _V
52 sseq2 3627 . . . . . 6 |- ( w = ( f ` x ) -> ( ( 1st ` x ) C_ w <-> ( 1st ` x ) C_ ( f ` x ) ) )
53 sseq1 3626 . . . . . 6 |- ( w = ( f ` x ) -> ( w C_ ( 2nd ` x ) <-> ( f ` x ) C_ ( 2nd ` x ) ) )
5452, 53anbi12d 747 . . . . 5 |- ( w = ( f ` x ) -> ( ( ( 1st ` x ) C_ w /\ w C_ ( 2nd ` x ) ) <-> ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) )
5551, 54axcc4dom 9263 . . . 4 |- ( ( S ~<_ om /\ A. x e. S E. w e. ( _I ` B ) ( ( 1st ` x ) C_ w /\ w C_ ( 2nd ` x ) ) ) -> E. f ( f : S --> ( _I ` B ) /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) )
5624, 50, 55syl2anc 693 . . 3 |- ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) -> E. f ( f : S --> ( _I ` B ) /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) )
5746ad2antrr 762 . . . . . . 7 |- ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) -> ( _I ` B ) = B )
5857feq3d 6032 . . . . . 6 |- ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) -> ( f : S --> ( _I ` B ) <-> f : S --> B ) )
5958anbi1d 741 . . . . 5 |- ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) -> ( ( f : S --> ( _I ` B ) /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) <-> ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) )
60 2ndctop 21250 . . . . . . . . . . . 12 |- ( J e. 2ndc -> J e. Top )
6160adantl 482 . . . . . . . . . . 11 |- ( ( B e. TopBases /\ J e. 2ndc ) -> J e. Top )
6261ad2antrr 762 . . . . . . . . . 10 |- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> J e. Top )
63 frn 6053 . . . . . . . . . . . 12 |- ( f : S --> B -> ran f C_ B )
6463ad2antrl 764 . . . . . . . . . . 11 |- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> ran f C_ B )
65 bastg 20770 . . . . . . . . . . . . 13 |- ( B e. TopBases -> B C_ ( topGen ` B ) )
6665ad3antrrr 766 . . . . . . . . . . . 12 |- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> B C_ ( topGen ` B ) )
67 2ndcctbss.2 . . . . . . . . . . . 12 |- J = ( topGen ` B )
6866, 67syl6sseqr 3652 . . . . . . . . . . 11 |- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> B C_ J )
6964, 68sstrd 3613 . . . . . . . . . 10 |- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> ran f C_ J )
70 simprrl 804 . . . . . . . . . . . . . . 15 |- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) -> o e. J )
71 simprr 796 . . . . . . . . . . . . . . . 16 |- ( ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) -> ( topGen ` c ) = J )
7271ad2antlr 763 . . . . . . . . . . . . . . 15 |- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) -> ( topGen ` c ) = J )
7370, 72eleqtrrd 2704 . . . . . . . . . . . . . 14 |- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) -> o e. ( topGen ` c ) )
74 simprrr 805 . . . . . . . . . . . . . 14 |- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) -> t e. o )
75 tg2 20769 . . . . . . . . . . . . . 14 |- ( ( o e. ( topGen ` c ) /\ t e. o ) -> E. d e. c ( t e. d /\ d C_ o ) )
7673, 74, 75syl2anc 693 . . . . . . . . . . . . 13 |- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) -> E. d e. c ( t e. d /\ d C_ o ) )
77 bastg 20770 . . . . . . . . . . . . . . . . . . 19 |- ( c e. TopBases -> c C_ ( topGen ` c ) )
7877ad2antrl 764 . . . . . . . . . . . . . . . . . 18 |- ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) -> c C_ ( topGen ` c ) )
7978ad2antrr 762 . . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) -> c C_ ( topGen ` c ) )
8067eqeq2i 2634 . . . . . . . . . . . . . . . . . . . . 21 |- ( ( topGen ` c ) = J <-> ( topGen ` c ) = ( topGen ` B ) )
8180biimpi 206 . . . . . . . . . . . . . . . . . . . 20 |- ( ( topGen ` c ) = J -> ( topGen ` c ) = ( topGen ` B ) )
8281adantl 482 . . . . . . . . . . . . . . . . . . 19 |- ( ( c ~<_ om /\ ( topGen ` c ) = J ) -> ( topGen ` c ) = ( topGen ` B ) )
8382ad2antll 765 . . . . . . . . . . . . . . . . . 18 |- ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) -> ( topGen ` c ) = ( topGen ` B ) )
8483ad2antrr 762 . . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) -> ( topGen ` c ) = ( topGen ` B ) )
8579, 84sseqtrd 3641 . . . . . . . . . . . . . . . 16 |- ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) -> c C_ ( topGen ` B ) )
86 simprl 794 . . . . . . . . . . . . . . . 16 |- ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) -> d e. c )
8785, 86sseldd 3604 . . . . . . . . . . . . . . 15 |- ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) -> d e. ( topGen ` B ) )
88 simprrl 804 . . . . . . . . . . . . . . 15 |- ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) -> t e. d )
89 tg2 20769 . . . . . . . . . . . . . . 15 |- ( ( d e. ( topGen ` B ) /\ t e. d ) -> E. m e. B ( t e. m /\ m C_ d ) )
9087, 88, 89syl2anc 693 . . . . . . . . . . . . . 14 |- ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) -> E. m e. B ( t e. m /\ m C_ d ) )
9165ad3antrrr 766 . . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) -> B C_ ( topGen ` B ) )
9291ad2antrr 762 . . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) -> B C_ ( topGen ` B ) )
9372ad2antrr 762 . . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) -> ( topGen ` c ) = J )
9493, 67syl6req 2673 . . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) -> ( topGen ` B ) = ( topGen ` c ) )
9592, 94sseqtrd 3641 . . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) -> B C_ ( topGen ` c ) )
96 simprl 794 . . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) -> m e. B )
9795, 96sseldd 3604 . . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) -> m e. ( topGen ` c ) )
98 simprrl 804 . . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) -> t e. m )
99 tg2 20769 . . . . . . . . . . . . . . . 16 |- ( ( m e. ( topGen ` c ) /\ t e. m ) -> E. n e. c ( t e. n /\ n C_ m ) )
10097, 98, 99syl2anc 693 . . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) -> E. n e. c ( t e. n /\ n C_ m ) )
101 ffn 6045 . . . . . . . . . . . . . . . . . . . 20 |- ( f : S --> B -> f Fn S )
102101ad2antrr 762 . . . . . . . . . . . . . . . . . . 19 |- ( ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) -> f Fn S )
103102ad2antlr 763 . . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) -> f Fn S )
104103ad2antrr 762 . . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> f Fn S )
105 simprl 794 . . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> n e. c )
10686ad2antrr 762 . . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> d e. c )
107 simplrl 800 . . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> m e. B )
108 simprrr 805 . . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> n C_ m )
109 simprr 796 . . . . . . . . . . . . . . . . . . . 20 |- ( ( m e. B /\ ( t e. m /\ m C_ d ) ) -> m C_ d )
110109ad2antlr 763 . . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> m C_ d )
111 sseq2 3627 . . . . . . . . . . . . . . . . . . . . 21 |- ( w = m -> ( n C_ w <-> n C_ m ) )
112 sseq1 3626 . . . . . . . . . . . . . . . . . . . . 21 |- ( w = m -> ( w C_ d <-> m C_ d ) )
113111, 112anbi12d 747 . . . . . . . . . . . . . . . . . . . 20 |- ( w = m -> ( ( n C_ w /\ w C_ d ) <-> ( n C_ m /\ m C_ d ) ) )
114113rspcev 3309 . . . . . . . . . . . . . . . . . . 19 |- ( ( m e. B /\ ( n C_ m /\ m C_ d ) ) -> E. w e. B ( n C_ w /\ w C_ d ) )
115107, 108, 110, 114syl12anc 1324 . . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> E. w e. B ( n C_ w /\ w C_ d ) )
116 df-br 4654 . . . . . . . . . . . . . . . . . . 19 |- ( n S d <-> <. n , d >. e. S )
117 vex 3203 . . . . . . . . . . . . . . . . . . . 20 |- n e. _V
118 vex 3203 . . . . . . . . . . . . . . . . . . . 20 |- d e. _V
119 simpl 473 . . . . . . . . . . . . . . . . . . . . . 22 |- ( ( u = n /\ v = d ) -> u = n )
120119eleq1d 2686 . . . . . . . . . . . . . . . . . . . . 21 |- ( ( u = n /\ v = d ) -> ( u e. c <-> n e. c ) )
121 simpr 477 . . . . . . . . . . . . . . . . . . . . . 22 |- ( ( u = n /\ v = d ) -> v = d )
122121eleq1d 2686 . . . . . . . . . . . . . . . . . . . . 21 |- ( ( u = n /\ v = d ) -> ( v e. c <-> d e. c ) )
123 sseq1 3626 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( u = n -> ( u C_ w <-> n C_ w ) )
124 sseq2 3627 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( v = d -> ( w C_ v <-> w C_ d ) )
125123, 124bi2anan9 917 . . . . . . . . . . . . . . . . . . . . . 22 |- ( ( u = n /\ v = d ) -> ( ( u C_ w /\ w C_ v ) <-> ( n C_ w /\ w C_ d ) ) )
126125rexbidv 3052 . . . . . . . . . . . . . . . . . . . . 21 |- ( ( u = n /\ v = d ) -> ( E. w e. B ( u C_ w /\ w C_ v ) <-> E. w e. B ( n C_ w /\ w C_ d ) ) )
127120, 122, 1263anbi123d 1399 . . . . . . . . . . . . . . . . . . . 20 |- ( ( u = n /\ v = d ) -> ( ( u e. c /\ v e. c /\ E. w e. B ( u C_ w /\ w C_ v ) ) <-> ( n e. c /\ d e. c /\ E. w e. B ( n C_ w /\ w C_ d ) ) ) )
128117, 118, 127, 8braba 4992 . . . . . . . . . . . . . . . . . . 19 |- ( n S d <-> ( n e. c /\ d e. c /\ E. w e. B ( n C_ w /\ w C_ d ) ) )
129116, 128bitr3i 266 . . . . . . . . . . . . . . . . . 18 |- ( <. n , d >. e. S <-> ( n e. c /\ d e. c /\ E. w e. B ( n C_ w /\ w C_ d ) ) )
130105, 106, 115, 129syl3anbrc 1246 . . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> <. n , d >. e. S )
131 fnfvelrn 6356 . . . . . . . . . . . . . . . . 17 |- ( ( f Fn S /\ <. n , d >. e. S ) -> ( f ` <. n , d >. ) e. ran f )
132104, 130, 131syl2anc 693 . . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> ( f ` <. n , d >. ) e. ran f )
133 simprl 794 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> n e. c )
134 simplll 798 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> d e. c )
135 simplrl 800 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> m e. B )
136 simprrr 805 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> n C_ m )
137109ad2antlr 763 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> m C_ d )
138135, 136, 137, 114syl12anc 1324 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> E. w e. B ( n C_ w /\ w C_ d ) )
139133, 134, 138, 129syl3anbrc 1246 . . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> <. n , d >. e. S )
140 fveq2 6191 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( x = <. n , d >. -> ( 1st ` x ) = ( 1st ` <. n , d >. ) )
141 fveq2 6191 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( x = <. n , d >. -> ( f ` x ) = ( f ` <. n , d >. ) )
142140, 141sseq12d 3634 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ( x = <. n , d >. -> ( ( 1st ` x ) C_ ( f ` x ) <-> ( 1st ` <. n , d >. ) C_ ( f ` <. n , d >. ) ) )
143 fveq2 6191 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( x = <. n , d >. -> ( 2nd ` x ) = ( 2nd ` <. n , d >. ) )
144141, 143sseq12d 3634 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ( x = <. n , d >. -> ( ( f ` x ) C_ ( 2nd ` x ) <-> ( f ` <. n , d >. ) C_ ( 2nd ` <. n , d >. ) ) )
145142, 144anbi12d 747 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( x = <. n , d >. -> ( ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) <-> ( ( 1st ` <. n , d >. ) C_ ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ ( 2nd ` <. n , d >. ) ) ) )
146145rspcv 3305 . . . . . . . . . . . . . . . . . . . . . 22 |- ( <. n , d >. e. S -> ( A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) -> ( ( 1st ` <. n , d >. ) C_ ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ ( 2nd ` <. n , d >. ) ) ) )
147139, 146syl 17 . . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> ( A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) -> ( ( 1st ` <. n , d >. ) C_ ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ ( 2nd ` <. n , d >. ) ) ) )
148117, 118op1st 7176 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ( 1st ` <. n , d >. ) = n
149148sseq1i 3629 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( 1st ` <. n , d >. ) C_ ( f ` <. n , d >. ) <-> n C_ ( f ` <. n , d >. ) )
150117, 118op2nd 7177 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ( 2nd ` <. n , d >. ) = d
151150sseq2i 3630 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( f ` <. n , d >. ) C_ ( 2nd ` <. n , d >. ) <-> ( f ` <. n , d >. ) C_ d )
152149, 151anbi12i 733 . . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( 1st ` <. n , d >. ) C_ ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ ( 2nd ` <. n , d >. ) ) <-> ( n C_ ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ d ) )
153 simprl 794 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) /\ ( n C_ ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ d ) ) -> n C_ ( f ` <. n , d >. ) )
154 simprl 794 . . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( n e. c /\ ( t e. n /\ n C_ m ) ) -> t e. n )
155154ad2antlr 763 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) /\ ( n C_ ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ d ) ) -> t e. n )
156153, 155sseldd 3604 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) /\ ( n C_ ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ d ) ) -> t e. ( f ` <. n , d >. ) )
157 simprr 796 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) /\ ( n C_ ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ d ) ) -> ( f ` <. n , d >. ) C_ d )
158 simplrr 801 . . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) -> d C_ o )
159158ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) /\ ( n C_ ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ d ) ) -> d C_ o )
160157, 159sstrd 3613 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) /\ ( n C_ ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ d ) ) -> ( f ` <. n , d >. ) C_ o )
161156, 160jca 554 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) /\ ( n C_ ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ d ) ) -> ( t e. ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ o ) )
162161ex 450 . . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> ( ( n C_ ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ d ) -> ( t e. ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ o ) ) )
163152, 162syl5bi 232 . . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> ( ( ( 1st ` <. n , d >. ) C_ ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ ( 2nd ` <. n , d >. ) ) -> ( t e. ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ o ) ) )
164147, 163syldc 48 . . . . . . . . . . . . . . . . . . . 20 |- ( A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) -> ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> ( t e. ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ o ) ) )
165164exp4c 636 . . . . . . . . . . . . . . . . . . 19 |- ( A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) -> ( ( d e. c /\ ( t e. d /\ d C_ o ) ) -> ( ( m e. B /\ ( t e. m /\ m C_ d ) ) -> ( ( n e. c /\ ( t e. n /\ n C_ m ) ) -> ( t e. ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ o ) ) ) ) )
166165ad2antlr 763 . . . . . . . . . . . . . . . . . 18 |- ( ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) -> ( ( d e. c /\ ( t e. d /\ d C_ o ) ) -> ( ( m e. B /\ ( t e. m /\ m C_ d ) ) -> ( ( n e. c /\ ( t e. n /\ n C_ m ) ) -> ( t e. ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ o ) ) ) ) )
167166adantl 482 . . . . . . . . . . . . . . . . 17 |- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) -> ( ( d e. c /\ ( t e. d /\ d C_ o ) ) -> ( ( m e. B /\ ( t e. m /\ m C_ d ) ) -> ( ( n e. c /\ ( t e. n /\ n C_ m ) ) -> ( t e. ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ o ) ) ) ) )
168167imp41 619 . . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> ( t e. ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ o ) )
169 eleq2 2690 . . . . . . . . . . . . . . . . . 18 |- ( b = ( f ` <. n , d >. ) -> ( t e. b <-> t e. ( f ` <. n , d >. ) ) )
170 sseq1 3626 . . . . . . . . . . . . . . . . . 18 |- ( b = ( f ` <. n , d >. ) -> ( b C_ o <-> ( f ` <. n , d >. ) C_ o ) )
171169, 170anbi12d 747 . . . . . . . . . . . . . . . . 17 |- ( b = ( f ` <. n , d >. ) -> ( ( t e. b /\ b C_ o ) <-> ( t e. ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ o ) ) )
172171rspcev 3309 . . . . . . . . . . . . . . . 16 |- ( ( ( f ` <. n , d >. ) e. ran f /\ ( t e. ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ o ) ) -> E. b e. ran f ( t e. b /\ b C_ o ) )
173132, 168, 172syl2anc 693 . . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> E. b e. ran f ( t e. b /\ b C_ o ) )
174100, 173rexlimddv 3035 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) -> E. b e. ran f ( t e. b /\ b C_ o ) )
17590, 174rexlimddv 3035 . . . . . . . . . . . . 13 |- ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) -> E. b e. ran f ( t e. b /\ b C_ o ) )
17676, 175rexlimddv 3035 . . . . . . . . . . . 12 |- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) -> E. b e. ran f ( t e. b /\ b C_ o ) )
177176expr 643 . . . . . . . . . . 11 |- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> ( ( o e. J /\ t e. o ) -> E. b e. ran f ( t e. b /\ b C_ o ) ) )
178177ralrimivv 2970 . . . . . . . . . 10 |- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> A. o e. J A. t e. o E. b e. ran f ( t e. b /\ b C_ o ) )
179 basgen2 20793 . . . . . . . . . 10 |- ( ( J e. Top /\ ran f C_ J /\ A. o e. J A. t e. o E. b e. ran f ( t e. b /\ b C_ o ) ) -> ( topGen ` ran f ) = J )
18062, 69, 178, 179syl3anc 1326 . . . . . . . . 9 |- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> ( topGen ` ran f ) = J )
181180, 62eqeltrd 2701 . . . . . . . 8 |- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> ( topGen ` ran f ) e. Top )
182 tgclb 20774 . . . . . . . 8 |- ( ran f e. TopBases <-> ( topGen ` ran f ) e. Top )
183181, 182sylibr 224 . . . . . . 7 |- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> ran f e. TopBases )
184 omelon 8543 . . . . . . . . . 10 |- om e. On
18524adantr 481 . . . . . . . . . 10 |- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> S ~<_ om )
186 ondomen 8860 . . . . . . . . . 10 |- ( ( om e. On /\ S ~<_ om ) -> S e. dom card )
187184, 185, 186sylancr 695 . . . . . . . . 9 |- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> S e. dom card )
188101ad2antrl 764 . . . . . . . . . 10 |- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> f Fn S )
189 dffn4 6121 . . . . . . . . . 10 |- ( f Fn S <-> f : S -onto-> ran f )
190188, 189sylib 208 . . . . . . . . 9 |- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> f : S -onto-> ran f )
191 fodomnum 8880 . . . . . . . . 9 |- ( S e. dom card -> ( f : S -onto-> ran f -> ran f ~<_ S ) )
192187, 190, 191sylc 65 . . . . . . . 8 |- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> ran f ~<_ S )
193 domtr 8009 . . . . . . . 8 |- ( ( ran f ~<_ S /\ S ~<_ om ) -> ran f ~<_ om )
194192, 185, 193syl2anc 693 . . . . . . 7 |- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> ran f ~<_ om )
195180eqcomd 2628 . . . . . . 7 |- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> J = ( topGen ` ran f ) )
196 breq1 4656 . . . . . . . . 9 |- ( b = ran f -> ( b ~<_ om <-> ran f ~<_ om ) )
197 sseq1 3626 . . . . . . . . 9 |- ( b = ran f -> ( b C_ B <-> ran f C_ B ) )
198 fveq2 6191 . . . . . . . . . 10 |- ( b = ran f -> ( topGen ` b ) = ( topGen ` ran f ) )
199198eqeq2d 2632 . . . . . . . . 9 |- ( b = ran f -> ( J = ( topGen ` b ) <-> J = ( topGen ` ran f ) ) )
200196, 197, 1993anbi123d 1399 . . . . . . . 8 |- ( b = ran f -> ( ( b ~<_ om /\ b C_ B /\ J = ( topGen ` b ) ) <-> ( ran f ~<_ om /\ ran f C_ B /\ J = ( topGen ` ran f ) ) ) )
201200rspcev 3309 . . . . . . 7 |- ( ( ran f e. TopBases /\ ( ran f ~<_ om /\ ran f C_ B /\ J = ( topGen ` ran f ) ) ) -> E. b e. TopBases ( b ~<_ om /\ b C_ B /\ J = ( topGen ` b ) ) )
202183, 194, 64, 195, 201syl13anc 1328 . . . . . 6 |- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> E. b e. TopBases ( b ~<_ om /\ b C_ B /\ J = ( topGen ` b ) ) )
203202ex 450 . . . . 5 |- ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) -> ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) -> E. b e. TopBases ( b ~<_ om /\ b C_ B /\ J = ( topGen ` b ) ) ) )
20459, 203sylbid 230 . . . 4 |- ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) -> ( ( f : S --> ( _I ` B ) /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) -> E. b e. TopBases ( b ~<_ om /\ b C_ B /\ J = ( topGen ` b ) ) ) )
205204exlimdv 1861 . . 3 |- ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) -> ( E. f ( f : S --> ( _I ` B ) /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) -> E. b e. TopBases ( b ~<_ om /\ b C_ B /\ J = ( topGen ` b ) ) ) )
20656, 205mpd 15 . 2 |- ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ om /\ ( topGen ` c ) = J ) ) ) -> E. b e. TopBases ( b ~<_ om /\ b C_ B /\ J = ( topGen ` b ) ) )
2073, 206rexlimddv 3035 1 |- ( ( B e. TopBases /\ J e. 2ndc ) -> E. b e. TopBases ( b ~<_ om /\ b C_ B /\ J = ( topGen ` b ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   <.cop 4183   U.cuni 4436   class class class wbr 4653   {copab 4712   _I cid 5023   X. cxp 5112   dom cdm 5114   ran crn 5115   Rel wrel 5119   Oncon0 5723   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   ` cfv 5888   omcom 7065   1stc1st 7166   2ndc2nd 7167   ~~ cen 7952   ~<_ cdom 7953   cardccrd 8761   topGenctg 16098   Topctop 20698   TopBasesctb 20749   2ndcc2ndc 21241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cc 9257
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-card 8765  df-acn 8768  df-topgen 16104  df-top 20699  df-bases 20750  df-2ndc 21243
This theorem is referenced by: (None)
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