Theorem bnj1450 31118

Description: Technical lemma for bnj60 31130. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1450.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj1450.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj1450.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1450.4	|- ( ta <-> ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
bnj1450.5	|- D = { x e. A | -. E. f ta }
bnj1450.6	|- ( ps <-> ( R FrSe A /\ D =/= (/) ) )
bnj1450.7	|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
bnj1450.8	|- ( ta' <-> [. y / x ]. ta )
bnj1450.9	|- H = { f | E. y e. pred ( x , A , R ) ta' }
bnj1450.10	|- P = U. H
bnj1450.11	|- Z = <. x , ( P |` pred ( x , A , R ) ) >.
bnj1450.12	|- Q = ( P u. { <. x , ( G ` Z ) >. } )
bnj1450.13	|- W = <. z , ( Q |` pred ( z , A , R ) ) >.
bnj1450.14	|- E = ( { x } u. trCl ( x , A , R ) )
bnj1450.15	|- ( ch -> P Fn trCl ( x , A , R ) )
bnj1450.16	|- ( ch -> Q Fn ( { x } u. trCl ( x , A , R ) ) )
bnj1450.17	|- ( th <-> ( ch /\ z e. E ) )
bnj1450.18	|- ( et <-> ( th /\ z e. { x } ) )
bnj1450.19	|- ( ze <-> ( th /\ z e. trCl ( x , A , R ) ) )
bnj1450.20	|- ( rh <-> ( ze /\ f e. H /\ z e. dom f ) )
bnj1450.21	|- ( si <-> ( rh /\ y e. pred ( x , A , R ) /\ f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) )
bnj1450.22	|- ( ph <-> ( si /\ d e. B /\ f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) )
bnj1450.23	|- X = <. z , ( f |` pred ( z , A , R ) ) >.

Assertion

Ref	Expression
bnj1450	|- ( ze -> ( Q ` z ) = ( G ` W ) )

Distinct variable groups:   A, d, f, x, y, z   B, f   y, D   E, d, f, y   G, d, f, x, y, z   R, d, f, x, y, z   x, X   z, Y   ps, y

Allowed substitution hints:   ph( x, y, z, f, d)   ps( x, z, f, d)   ch( x, y, z, f, d)   th( x, y, z, f, d)   ta( x, y, z, f, d)   et( x, y, z, f, d)   ze( x, y, z, f, d)   si( x, y, z, f, d)   rh( x, y, z, f, d)   B( x, y, z, d)   C( x, y, z, f, d)   D( x, z, f, d)   P( x, y, z, f, d)   Q( x, y, z, f, d)   E( x, z)   H( x, y, z, f, d)   W( x, y, z, f, d)   X( y, z, f, d)   Y( x, y, f, d)   Z( x, y, z, f, d)   ta'( x, y, z, f, d)

Proof of Theorem bnj1450

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1450.19	. . . . . . . . 9 |- ( ze <-> ( th /\ z e. trCl ( x , A , R ) ) )
2	1	simprbi 480	. . . . . . . 8 |- ( ze -> z e. trCl ( x , A , R ) )
3		bnj1450.17	. . . . . . . . . 10 |- ( th <-> ( ch /\ z e. E ) )
4		bnj1450.15	. . . . . . . . . . 11 |- ( ch -> P Fn trCl ( x , A , R ) )
5		fndm 5990	. . . . . . . . . . 11 |- ( P Fn trCl ( x , A , R ) -> dom P = trCl ( x , A , R ) )
6	4, 5	syl 17	. . . . . . . . . 10 |- ( ch -> dom P = trCl ( x , A , R ) )
7	3, 6	bnj832 30828	. . . . . . . . 9 |- ( th -> dom P = trCl ( x , A , R ) )
8	1, 7	bnj832 30828	. . . . . . . 8 |- ( ze -> dom P = trCl ( x , A , R ) )
9	2, 8	eleqtrrd 2704	. . . . . . 7 |- ( ze -> z e. dom P )
10		bnj1450.10	. . . . . . . 8 |- P = U. H
11	10	dmeqi 5325	. . . . . . 7 |- dom P = dom U. H
12	9, 11	syl6eleq 2711	. . . . . 6 |- ( ze -> z e. dom U. H )
13		bnj1450.9	. . . . . . . 8 |- H = { f | E. y e. pred ( x , A , R ) ta' }
14	13	bnj1317 30892	. . . . . . 7 |- ( w e. H -> A. f w e. H )
15	14	bnj1400 30906	. . . . . 6 |- dom U. H = U_ f e. H dom f
16	12, 15	syl6eleq 2711	. . . . 5 |- ( ze -> z e. U_ f e. H dom f )
17	16	bnj1405 30907	. . . 4 |- ( ze -> E. f e. H z e. dom f )
18		bnj1450.20	. . . 4 |- ( rh <-> ( ze /\ f e. H /\ z e. dom f ) )
19		bnj1450.1	. . . . 5 |- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
20		bnj1450.2	. . . . 5 |- Y = <. x , ( f |` pred ( x , A , R ) ) >.
21		bnj1450.3	. . . . 5 |- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
22		bnj1450.4	. . . . 5 |- ( ta <-> ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
23		bnj1450.5	. . . . 5 |- D = { x e. A | -. E. f ta }
24		bnj1450.6	. . . . 5 |- ( ps <-> ( R FrSe A /\ D =/= (/) ) )
25		bnj1450.7	. . . . 5 |- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
26		bnj1450.8	. . . . 5 |- ( ta' <-> [. y / x ]. ta )
27		bnj1450.11	. . . . 5 |- Z = <. x , ( P |` pred ( x , A , R ) ) >.
28		bnj1450.12	. . . . 5 |- Q = ( P u. { <. x , ( G ` Z ) >. } )
29		bnj1450.13	. . . . 5 |- W = <. z , ( Q |` pred ( z , A , R ) ) >.
30		bnj1450.14	. . . . 5 |- E = ( { x } u. trCl ( x , A , R ) )
31		bnj1450.16	. . . . 5 |- ( ch -> Q Fn ( { x } u. trCl ( x , A , R ) ) )
32		bnj1450.18	. . . . 5 |- ( et <-> ( th /\ z e. { x } ) )
33	19, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29, 30, 4, 31, 3, 32, 1	bnj1449 31116	. . . 4 |- ( ze -> A. f ze )
34	17, 18, 33	bnj1521 30921	. . 3 |- ( ze -> E. f rh )
35	13	bnj1436 30910	. . . . . . . . . 10 |- ( f e. H -> E. y e. pred ( x , A , R ) ta' )
36	18, 35	bnj836 30830	. . . . . . . . 9 |- ( rh -> E. y e. pred ( x , A , R ) ta' )
37	19, 20, 21, 22, 26	bnj1373 31098	. . . . . . . . . 10 |- ( ta' <-> ( f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) )
38	37	rexbii 3041	. . . . . . . . 9 |- ( E. y e. pred ( x , A , R ) ta' <-> E. y e. pred ( x , A , R ) ( f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) )
39	36, 38	sylib 208	. . . . . . . 8 |- ( rh -> E. y e. pred ( x , A , R ) ( f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) )
40	39	bnj1196 30865	. . . . . . 7 |- ( rh -> E. y ( y e. pred ( x , A , R ) /\ ( f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) ) )
41		3anass 1042	. . . . . . 7 |- ( ( y e. pred ( x , A , R ) /\ f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) <-> ( y e. pred ( x , A , R ) /\ ( f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) ) )
42	40, 41	bnj1198 30866	. . . . . 6 |- ( rh -> E. y ( y e. pred ( x , A , R ) /\ f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) )
43		bnj1450.21	. . . . . . 7 |- ( si <-> ( rh /\ y e. pred ( x , A , R ) /\ f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) )
44		bnj252 30769	. . . . . . 7 |- ( ( rh /\ y e. pred ( x , A , R ) /\ f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) <-> ( rh /\ ( y e. pred ( x , A , R ) /\ f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) ) )
45	43, 44	bitri 264	. . . . . 6 |- ( si <-> ( rh /\ ( y e. pred ( x , A , R ) /\ f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) ) )
46	19, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29, 30, 4, 31, 3, 32, 1, 18	bnj1444 31111	. . . . . 6 |- ( rh -> A. y rh )
47	42, 45, 46	bnj1340 30894	. . . . 5 |- ( rh -> E. y si )
48	21	bnj1436 30910	. . . . . . . . . . 11 |- ( f e. C -> E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) )
49	43, 48	bnj771 30834	. . . . . . . . . 10 |- ( si -> E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) )
50	49	bnj1196 30865	. . . . . . . . 9 |- ( si -> E. d ( d e. B /\ ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) ) )
51		3anass 1042	. . . . . . . . 9 |- ( ( d e. B /\ f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) <-> ( d e. B /\ ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) ) )
52	50, 51	bnj1198 30866	. . . . . . . 8 |- ( si -> E. d ( d e. B /\ f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) )
53		bnj1450.22	. . . . . . . . 9 |- ( ph <-> ( si /\ d e. B /\ f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) )
54		bnj252 30769	. . . . . . . . 9 |- ( ( si /\ d e. B /\ f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) <-> ( si /\ ( d e. B /\ f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) ) )
55	53, 54	bitri 264	. . . . . . . 8 |- ( ph <-> ( si /\ ( d e. B /\ f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) ) )
56		bnj1450.23	. . . . . . . . 9 |- X = <. z , ( f |` pred ( z , A , R ) ) >.
57	19, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29, 30, 4, 31, 3, 32, 1, 18, 43, 53, 56	bnj1445 31112	. . . . . . . 8 |- ( si -> A. d si )
58	52, 55, 57	bnj1340 30894	. . . . . . 7 |- ( si -> E. d ph )
59	53	bnj1254 30880	. . . . . . . . . 10 |- ( ph -> A. x e. d ( f ` x ) = ( G ` Y ) )
60		fveq2 6191	. . . . . . . . . . . 12 |- ( x = z -> ( f ` x ) = ( f ` z ) )
61		id 22	. . . . . . . . . . . . . . 15 |- ( x = z -> x = z )
62		bnj602 30985	. . . . . . . . . . . . . . . 16 |- ( x = z -> pred ( x , A , R ) = pred ( z , A , R ) )
63	62	reseq2d 5396	. . . . . . . . . . . . . . 15 |- ( x = z -> ( f |` pred ( x , A , R ) ) = ( f |` pred ( z , A , R ) ) )
64	61, 63	opeq12d 4410	. . . . . . . . . . . . . 14 |- ( x = z -> <. x , ( f |` pred ( x , A , R ) ) >. = <. z , ( f |` pred ( z , A , R ) ) >. )
65	64, 20, 56	3eqtr4g 2681	. . . . . . . . . . . . 13 |- ( x = z -> Y = X )
66	65	fveq2d 6195	. . . . . . . . . . . 12 |- ( x = z -> ( G ` Y ) = ( G ` X ) )
67	60, 66	eqeq12d 2637	. . . . . . . . . . 11 |- ( x = z -> ( ( f ` x ) = ( G ` Y ) <-> ( f ` z ) = ( G ` X ) ) )
68	67	cbvralv 3171	. . . . . . . . . 10 |- ( A. x e. d ( f ` x ) = ( G ` Y ) <-> A. z e. d ( f ` z ) = ( G ` X ) )
69	59, 68	sylib 208	. . . . . . . . 9 |- ( ph -> A. z e. d ( f ` z ) = ( G ` X ) )
70	18	simp3bi 1078	. . . . . . . . . . . 12 |- ( rh -> z e. dom f )
71	43, 70	bnj769 30832	. . . . . . . . . . 11 |- ( si -> z e. dom f )
72	53, 71	bnj769 30832	. . . . . . . . . 10 |- ( ph -> z e. dom f )
73		fndm 5990	. . . . . . . . . . 11 |- ( f Fn d -> dom f = d )
74	53, 73	bnj771 30834	. . . . . . . . . 10 |- ( ph -> dom f = d )
75	72, 74	eleqtrd 2703	. . . . . . . . 9 |- ( ph -> z e. d )
76	69, 75	bnj1294 30888	. . . . . . . 8 |- ( ph -> ( f ` z ) = ( G ` X ) )
77	31	bnj930 30840	. . . . . . . . . . . . . 14 |- ( ch -> Fun Q )
78	3, 77	bnj832 30828	. . . . . . . . . . . . 13 |- ( th -> Fun Q )
79	1, 78	bnj832 30828	. . . . . . . . . . . 12 |- ( ze -> Fun Q )
80	18, 79	bnj835 30829	. . . . . . . . . . 11 |- ( rh -> Fun Q )
81	43, 80	bnj769 30832	. . . . . . . . . 10 |- ( si -> Fun Q )
82	53, 81	bnj769 30832	. . . . . . . . 9 |- ( ph -> Fun Q )
83	18	simp2bi 1077	. . . . . . . . . . . 12 |- ( rh -> f e. H )
84	43, 83	bnj769 30832	. . . . . . . . . . 11 |- ( si -> f e. H )
85	53, 84	bnj769 30832	. . . . . . . . . 10 |- ( ph -> f e. H )
86		elssuni 4467	. . . . . . . . . . 11 |- ( f e. H -> f C_ U. H )
87	86, 10	syl6sseqr 3652	. . . . . . . . . 10 |- ( f e. H -> f C_ P )
88		ssun3 3778	. . . . . . . . . . 11 |- ( f C_ P -> f C_ ( P u. { <. x , ( G ` Z ) >. } ) )
89	88, 28	syl6sseqr 3652	. . . . . . . . . 10 |- ( f C_ P -> f C_ Q )
90	85, 87, 89	3syl 18	. . . . . . . . 9 |- ( ph -> f C_ Q )
91	82, 90, 72	bnj1502 30918	. . . . . . . 8 |- ( ph -> ( Q ` z ) = ( f ` z ) )
92	19	bnj1517 30920	. . . . . . . . . . . . . . . 16 |- ( d e. B -> A. x e. d pred ( x , A , R ) C_ d )
93	53, 92	bnj770 30833	. . . . . . . . . . . . . . 15 |- ( ph -> A. x e. d pred ( x , A , R ) C_ d )
94	62	sseq1d 3632	. . . . . . . . . . . . . . . 16 |- ( x = z -> ( pred ( x , A , R ) C_ d <-> pred ( z , A , R ) C_ d ) )
95	94	cbvralv 3171	. . . . . . . . . . . . . . 15 |- ( A. x e. d pred ( x , A , R ) C_ d <-> A. z e. d pred ( z , A , R ) C_ d )
96	93, 95	sylib 208	. . . . . . . . . . . . . 14 |- ( ph -> A. z e. d pred ( z , A , R ) C_ d )
97	96, 75	bnj1294 30888	. . . . . . . . . . . . 13 |- ( ph -> pred ( z , A , R ) C_ d )
98	97, 74	sseqtr4d 3642	. . . . . . . . . . . 12 |- ( ph -> pred ( z , A , R ) C_ dom f )
99	82, 90, 98	bnj1503 30919	. . . . . . . . . . 11 |- ( ph -> ( Q |` pred ( z , A , R ) ) = ( f |` pred ( z , A , R ) ) )
100	99	opeq2d 4409	. . . . . . . . . 10 |- ( ph -> <. z , ( Q |` pred ( z , A , R ) ) >. = <. z , ( f |` pred ( z , A , R ) ) >. )
101	100, 29, 56	3eqtr4g 2681	. . . . . . . . 9 |- ( ph -> W = X )
102	101	fveq2d 6195	. . . . . . . 8 |- ( ph -> ( G ` W ) = ( G ` X ) )
103	76, 91, 102	3eqtr4d 2666	. . . . . . 7 |- ( ph -> ( Q ` z ) = ( G ` W ) )
104	58, 103	bnj593 30815	. . . . . 6 |- ( si -> E. d ( Q ` z ) = ( G ` W ) )
105	19, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29	bnj1446 31113	. . . . . 6 |- ( ( Q ` z ) = ( G ` W ) -> A. d ( Q ` z ) = ( G ` W ) )
106	104, 105	bnj1397 30905	. . . . 5 |- ( si -> ( Q ` z ) = ( G ` W ) )
107	47, 106	bnj593 30815	. . . 4 |- ( rh -> E. y ( Q ` z ) = ( G ` W ) )
108	19, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29	bnj1447 31114	. . . 4 |- ( ( Q ` z ) = ( G ` W ) -> A. y ( Q ` z ) = ( G ` W ) )
109	107, 108	bnj1397 30905	. . 3 |- ( rh -> ( Q ` z ) = ( G ` W ) )
110	34, 109	bnj593 30815	. 2 |- ( ze -> E. f ( Q ` z ) = ( G ` W ) )
111	19, 20, 21, 22, 23, 24, 25, 26, 13, 10, 27, 28, 29	bnj1448 31115	. 2 |- ( ( Q ` z ) = ( G ` W ) -> A. f ( Q ` z ) = ( G ` W ) )
112	110, 111	bnj1397 30905	1 |- ( ze -> ( Q ` z ) = ( G ` W ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   [.wsbc 3435   u. cun 3572   C_ wss 3574   (/)c0 3915   {csn 4177   <.cop 4183   U.cuni 4436   U_ciun 4520   class class class wbr 4653   dom cdm 5114   |` cres 5116   Fun wfun 5882   Fn wfn 5883   ` cfv 5888   /\ w-bnj17 30752   predc-bnj14 30754   FrSe w-bnj15 30758   trClc-bnj18 30760

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-bnj17 30753  df-bnj14 30755  df-bnj18 30761

This theorem is referenced by:  bnj1423  31119