Theorem bnj1523 31139

Description: Technical lemma for bnj1522 31140. 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
bnj1523.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj1523.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj1523.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1523.4	|- F = U. C
bnj1523.5	|- ( ph <-> ( R FrSe A /\ H Fn A /\ A. x e. A ( H ` x ) = ( G ` <. x , ( H |` pred ( x , A , R ) ) >. ) ) )
bnj1523.6	|- ( ps <-> ( ph /\ F =/= H ) )
bnj1523.7	|- ( ch <-> ( ps /\ x e. A /\ ( F ` x ) =/= ( H ` x ) ) )
bnj1523.8	|- D = { x e. A | ( F ` x ) =/= ( H ` x ) }
bnj1523.9	|- ( th <-> ( ch /\ y e. D /\ A. z e. D -. z R y ) )

Assertion

Ref	Expression
bnj1523	|- ( ( R FrSe A /\ H Fn A /\ A. x e. A ( H ` x ) = ( G ` <. x , ( H |` pred ( x , A , R ) ) >. ) ) -> F = H )

Distinct variable groups:   A, d, f, x   y, A, z, x   B, f   y, D, z   y, F, z   G, d, f, x   y, G   x, H, y, z   R, d, f, x   y, R, z   Y, d   ch, y

Allowed substitution hints:   ph( x, y, z, f, d)   ps( x, y, z, f, d)   ch( x, z, f, d)   th( x, y, z, f, d)   B( x, y, z, d)   C( x, y, z, f, d)   D( x, f, d)   F( x, f, d)   G( z)   H( f, d)   Y( x, y, z, f)

Proof of Theorem bnj1523

Dummy variables v w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1523.5	. 2 |- ( ph <-> ( R FrSe A /\ H Fn A /\ A. x e. A ( H ` x ) = ( G ` <. x , ( H |` pred ( x , A , R ) ) >. ) ) )
2		bnj1523.6	. . 3 |- ( ps <-> ( ph /\ F =/= H ) )
3		bnj1523.9	. . . . . . . . . . . . 13 |- ( th <-> ( ch /\ y e. D /\ A. z e. D -. z R y ) )
4		bnj1523.7	. . . . . . . . . . . . . 14 |- ( ch <-> ( ps /\ x e. A /\ ( F ` x ) =/= ( H ` x ) ) )
5		bnj1523.1	. . . . . . . . . . . . . . . . 17 |- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
6		bnj1523.2	. . . . . . . . . . . . . . . . 17 |- Y = <. x , ( f |` pred ( x , A , R ) ) >.
7		bnj1523.3	. . . . . . . . . . . . . . . . 17 |- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
8		bnj1523.4	. . . . . . . . . . . . . . . . 17 |- F = U. C
9	5, 6, 7, 8	bnj60 31130	. . . . . . . . . . . . . . . 16 |- ( R FrSe A -> F Fn A )
10	1, 9	bnj835 30829	. . . . . . . . . . . . . . 15 |- ( ph -> F Fn A )
11	2, 10	bnj832 30828	. . . . . . . . . . . . . 14 |- ( ps -> F Fn A )
12	4, 11	bnj835 30829	. . . . . . . . . . . . 13 |- ( ch -> F Fn A )
13	3, 12	bnj835 30829	. . . . . . . . . . . 12 |- ( th -> F Fn A )
14	1	simp2bi 1077	. . . . . . . . . . . . . . 15 |- ( ph -> H Fn A )
15	2, 14	bnj832 30828	. . . . . . . . . . . . . 14 |- ( ps -> H Fn A )
16	4, 15	bnj835 30829	. . . . . . . . . . . . 13 |- ( ch -> H Fn A )
17	3, 16	bnj835 30829	. . . . . . . . . . . 12 |- ( th -> H Fn A )
18		bnj213 30952	. . . . . . . . . . . . 13 |- pred ( y , A , R ) C_ A
19	18	a1i 11	. . . . . . . . . . . 12 |- ( th -> pred ( y , A , R ) C_ A )
20	3	simp3bi 1078	. . . . . . . . . . . . . . . . 17 |- ( th -> A. z e. D -. z R y )
21	20	bnj1211 30868	. . . . . . . . . . . . . . . 16 |- ( th -> A. z ( z e. D -> -. z R y ) )
22		con2b 349	. . . . . . . . . . . . . . . . 17 |- ( ( z e. D -> -. z R y ) <-> ( z R y -> -. z e. D ) )
23	22	albii 1747	. . . . . . . . . . . . . . . 16 |- ( A. z ( z e. D -> -. z R y ) <-> A. z ( z R y -> -. z e. D ) )
24	21, 23	sylib 208	. . . . . . . . . . . . . . 15 |- ( th -> A. z ( z R y -> -. z e. D ) )
25		bnj1418 31108	. . . . . . . . . . . . . . . . 17 |- ( z e. pred ( y , A , R ) -> z R y )
26	25	imim1i 63	. . . . . . . . . . . . . . . 16 |- ( ( z R y -> -. z e. D ) -> ( z e. pred ( y , A , R ) -> -. z e. D ) )
27	26	alimi 1739	. . . . . . . . . . . . . . 15 |- ( A. z ( z R y -> -. z e. D ) -> A. z ( z e. pred ( y , A , R ) -> -. z e. D ) )
28	24, 27	syl 17	. . . . . . . . . . . . . 14 |- ( th -> A. z ( z e. pred ( y , A , R ) -> -. z e. D ) )
29	28	bnj1142 30860	. . . . . . . . . . . . 13 |- ( th -> A. z e. pred ( y , A , R ) -. z e. D )
30		bnj1523.8	. . . . . . . . . . . . . 14 |- D = { x e. A | ( F ` x ) =/= ( H ` x ) }
31	5	bnj1309 31090	. . . . . . . . . . . . . . . . . . 19 |- ( w e. B -> A. x w e. B )
32	7, 31	bnj1307 31091	. . . . . . . . . . . . . . . . . 18 |- ( w e. C -> A. x w e. C )
33	32	nfcii 2755	. . . . . . . . . . . . . . . . 17 |- F/_ x C
34	33	nfuni 4442	. . . . . . . . . . . . . . . 16 |- F/_ x U. C
35	8, 34	nfcxfr 2762	. . . . . . . . . . . . . . 15 |- F/_ x F
36	35	nfcrii 2757	. . . . . . . . . . . . . 14 |- ( w e. F -> A. x w e. F )
37	30, 36	bnj1534 30923	. . . . . . . . . . . . 13 |- D = { z e. A | ( F ` z ) =/= ( H ` z ) }
38	29, 18, 37	bnj1533 30922	. . . . . . . . . . . 12 |- ( th -> A. z e. pred ( y , A , R ) ( F ` z ) = ( H ` z ) )
39	13, 17, 19, 38	bnj1536 30924	. . . . . . . . . . 11 |- ( th -> ( F |` pred ( y , A , R ) ) = ( H |` pred ( y , A , R ) ) )
40	39	opeq2d 4409	. . . . . . . . . 10 |- ( th -> <. y , ( F |` pred ( y , A , R ) ) >. = <. y , ( H |` pred ( y , A , R ) ) >. )
41	40	fveq2d 6195	. . . . . . . . 9 |- ( th -> ( G ` <. y , ( F |` pred ( y , A , R ) ) >. ) = ( G ` <. y , ( H |` pred ( y , A , R ) ) >. ) )
42	5, 6, 7, 8	bnj1500 31136	. . . . . . . . . . . . . . 15 |- ( R FrSe A -> A. x e. A ( F ` x ) = ( G ` <. x , ( F |` pred ( x , A , R ) ) >. ) )
43	1, 42	bnj835 30829	. . . . . . . . . . . . . 14 |- ( ph -> A. x e. A ( F ` x ) = ( G ` <. x , ( F |` pred ( x , A , R ) ) >. ) )
44	2, 43	bnj832 30828	. . . . . . . . . . . . 13 |- ( ps -> A. x e. A ( F ` x ) = ( G ` <. x , ( F |` pred ( x , A , R ) ) >. ) )
45	4, 44	bnj835 30829	. . . . . . . . . . . 12 |- ( ch -> A. x e. A ( F ` x ) = ( G ` <. x , ( F |` pred ( x , A , R ) ) >. ) )
46	45, 36	bnj1529 31138	. . . . . . . . . . 11 |- ( ch -> A. y e. A ( F ` y ) = ( G ` <. y , ( F |` pred ( y , A , R ) ) >. ) )
47	3, 46	bnj835 30829	. . . . . . . . . 10 |- ( th -> A. y e. A ( F ` y ) = ( G ` <. y , ( F |` pred ( y , A , R ) ) >. ) )
48	30	ssrab3 3688	. . . . . . . . . . 11 |- D C_ A
49	3	simp2bi 1077	. . . . . . . . . . 11 |- ( th -> y e. D )
50	48, 49	bnj1213 30869	. . . . . . . . . 10 |- ( th -> y e. A )
51	47, 50	bnj1294 30888	. . . . . . . . 9 |- ( th -> ( F ` y ) = ( G ` <. y , ( F |` pred ( y , A , R ) ) >. ) )
52	1	simp3bi 1078	. . . . . . . . . . . . . 14 |- ( ph -> A. x e. A ( H ` x ) = ( G ` <. x , ( H |` pred ( x , A , R ) ) >. ) )
53	2, 52	bnj832 30828	. . . . . . . . . . . . 13 |- ( ps -> A. x e. A ( H ` x ) = ( G ` <. x , ( H |` pred ( x , A , R ) ) >. ) )
54	4, 53	bnj835 30829	. . . . . . . . . . . 12 |- ( ch -> A. x e. A ( H ` x ) = ( G ` <. x , ( H |` pred ( x , A , R ) ) >. ) )
55		ax-5 1839	. . . . . . . . . . . 12 |- ( v e. H -> A. x v e. H )
56	54, 55	bnj1529 31138	. . . . . . . . . . 11 |- ( ch -> A. y e. A ( H ` y ) = ( G ` <. y , ( H |` pred ( y , A , R ) ) >. ) )
57	3, 56	bnj835 30829	. . . . . . . . . 10 |- ( th -> A. y e. A ( H ` y ) = ( G ` <. y , ( H |` pred ( y , A , R ) ) >. ) )
58	57, 50	bnj1294 30888	. . . . . . . . 9 |- ( th -> ( H ` y ) = ( G ` <. y , ( H |` pred ( y , A , R ) ) >. ) )
59	41, 51, 58	3eqtr4d 2666	. . . . . . . 8 |- ( th -> ( F ` y ) = ( H ` y ) )
60	30, 36	bnj1534 30923	. . . . . . . . . . 11 |- D = { y e. A | ( F ` y ) =/= ( H ` y ) }
61	60	bnj1538 30925	. . . . . . . . . 10 |- ( y e. D -> ( F ` y ) =/= ( H ` y ) )
62	3, 61	bnj836 30830	. . . . . . . . 9 |- ( th -> ( F ` y ) =/= ( H ` y ) )
63	62	neneqd 2799	. . . . . . . 8 |- ( th -> -. ( F ` y ) = ( H ` y ) )
64	59, 63	pm2.65i 185	. . . . . . 7 |- -. th
65	64	nex 1731	. . . . . 6 |- -. E. y th
66	1	simp1bi 1076	. . . . . . . . . 10 |- ( ph -> R FrSe A )
67	2, 66	bnj832 30828	. . . . . . . . 9 |- ( ps -> R FrSe A )
68	4, 67	bnj835 30829	. . . . . . . 8 |- ( ch -> R FrSe A )
69	48	a1i 11	. . . . . . . 8 |- ( ch -> D C_ A )
70	4	simp2bi 1077	. . . . . . . . . 10 |- ( ch -> x e. A )
71	4	simp3bi 1078	. . . . . . . . . 10 |- ( ch -> ( F ` x ) =/= ( H ` x ) )
72	30	rabeq2i 3197	. . . . . . . . . 10 |- ( x e. D <-> ( x e. A /\ ( F ` x ) =/= ( H ` x ) ) )
73	70, 71, 72	sylanbrc 698	. . . . . . . . 9 |- ( ch -> x e. D )
74		ne0i 3921	. . . . . . . . 9 |- ( x e. D -> D =/= (/) )
75	73, 74	syl 17	. . . . . . . 8 |- ( ch -> D =/= (/) )
76		bnj69 31078	. . . . . . . 8 |- ( ( R FrSe A /\ D C_ A /\ D =/= (/) ) -> E. y e. D A. z e. D -. z R y )
77	68, 69, 75, 76	syl3anc 1326	. . . . . . 7 |- ( ch -> E. y e. D A. z e. D -. z R y )
78	77, 3	bnj1209 30867	. . . . . 6 |- ( ch -> E. y th )
79	65, 78	mto 188	. . . . 5 |- -. ch
80	79	nex 1731	. . . 4 |- -. E. x ch
81	2	simprbi 480	. . . . . 6 |- ( ps -> F =/= H )
82	11, 15, 81, 36	bnj1542 30927	. . . . 5 |- ( ps -> E. x e. A ( F ` x ) =/= ( H ` x ) )
83	5, 6, 7, 8, 1, 2	bnj1525 31137	. . . . 5 |- ( ps -> A. x ps )
84	82, 4, 83	bnj1521 30921	. . . 4 |- ( ps -> E. x ch )
85	80, 84	mto 188	. . 3 |- -. ps
86	2, 85	bnj1541 30926	. 2 |- ( ph -> F = H )
87	1, 86	sylbir 225	1 |- ( ( R FrSe A /\ H Fn A /\ A. x e. A ( H ` x ) = ( G ` <. x , ( H |` pred ( x , A , R ) ) >. ) ) -> F = H )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   C_ wss 3574   (/)c0 3915   <.cop 4183   U.cuni 4436   class class class wbr 4653   |` cres 5116   Fn wfn 5883   ` cfv 5888   predc-bnj14 30754   FrSe w-bnj15 30758

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-reg 8497  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-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-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-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-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-om 7066  df-1o 7560  df-bnj17 30753  df-bnj14 30755  df-bnj13 30757  df-bnj15 30759  df-bnj18 30761  df-bnj19 30763

This theorem is referenced by:  bnj1522  31140