Theorem bnj1463 32069

Description: Technical lemma for bnj60 32076. 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
bnj1463.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj1463.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj1463.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1463.4	|- ( ta <-> ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
bnj1463.5	|- D = { x e. A | -. E. f ta }
bnj1463.6	|- ( ps <-> ( R FrSe A /\ D =/= (/) ) )
bnj1463.7	|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
bnj1463.8	|- ( ta' <-> [. y / x ]. ta )
bnj1463.9	|- H = { f | E. y e. pred ( x , A , R ) ta' }
bnj1463.10	|- P = U. H
bnj1463.11	|- Z = <. x , ( P |` pred ( x , A , R ) ) >.
bnj1463.12	|- Q = ( P u. { <. x , ( G ` Z ) >. } )
bnj1463.13	|- W = <. z , ( Q |` pred ( z , A , R ) ) >.
bnj1463.14	|- E = ( { x } u. trCl ( x , A , R ) )
bnj1463.15	|- ( ch -> Q e. _V )
bnj1463.16	|- ( ch -> A. z e. E ( Q ` z ) = ( G ` W ) )
bnj1463.17	|- ( ch -> Q Fn E )
bnj1463.18	|- ( ch -> E e. B )

Assertion

Ref	Expression
bnj1463	|- ( ch -> Q e. C )

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

Allowed substitution hints:   ps( x, y, z, f, d)   ch( x, y, z, f, d)   ta( x, y, z, f, d)   A( y, z)   B( x, y, z, d)   C( x, y, z, f, d)   D( x, y, z, f, d)   P( x, y, z, f, d)   Q( x, y, f, d)   R( y, z)   E( x, y, f)   G( y)   H( x, y, z, f, d)   W( 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 bnj1463

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1463.18	. . . . . . 7 |- ( ch -> E e. B )
2		elex 3000	. . . . . . 7 |- ( E e. B -> E e. _V )
3	1, 2	syl 16	. . . . . 6 |- ( ch -> E e. _V )
4		eleq1 2503	. . . . . . . 8 |- ( d = E -> ( d e. B <-> E e. B ) )
5		fneq2 5519	. . . . . . . . 9 |- ( d = E -> ( Q Fn d <-> Q Fn E ) )
6		raleq 2936	. . . . . . . . 9 |- ( d = E -> ( A. z e. d ( Q ` z ) = ( G ` W ) <-> A. z e. E ( Q ` z ) = ( G ` W ) ) )
7	5, 6	anbi12d 710	. . . . . . . 8 |- ( d = E -> ( ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) ) <-> ( Q Fn E /\ A. z e. E ( Q ` z ) = ( G ` W ) ) ) )
8	4, 7	anbi12d 710	. . . . . . 7 |- ( d = E -> ( ( d e. B /\ ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) ) ) <-> ( E e. B /\ ( Q Fn E /\ A. z e. E ( Q ` z ) = ( G ` W ) ) ) ) )
9		bnj1463.1	. . . . . . . . . . . 12 |- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
10	9	bnj1317 31838	. . . . . . . . . . 11 |- ( w e. B -> A. d w e. B )
11	10	nfcii 2579	. . . . . . . . . 10 |- F/_ d B
12	11	nfel2 2606	. . . . . . . . 9 |- F/ d E e. B
13		bnj1463.2	. . . . . . . . . . . . 13 |- Y = <. x , ( f |` pred ( x , A , R ) ) >.
14		bnj1463.3	. . . . . . . . . . . . 13 |- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
15		bnj1463.4	. . . . . . . . . . . . 13 |- ( ta <-> ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
16		bnj1463.5	. . . . . . . . . . . . 13 |- D = { x e. A | -. E. f ta }
17		bnj1463.6	. . . . . . . . . . . . 13 |- ( ps <-> ( R FrSe A /\ D =/= (/) ) )
18		bnj1463.7	. . . . . . . . . . . . 13 |- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
19		bnj1463.8	. . . . . . . . . . . . 13 |- ( ta' <-> [. y / x ]. ta )
20		bnj1463.9	. . . . . . . . . . . . 13 |- H = { f | E. y e. pred ( x , A , R ) ta' }
21		bnj1463.10	. . . . . . . . . . . . 13 |- P = U. H
22		bnj1463.11	. . . . . . . . . . . . 13 |- Z = <. x , ( P |` pred ( x , A , R ) ) >.
23		bnj1463.12	. . . . . . . . . . . . 13 |- Q = ( P u. { <. x , ( G ` Z ) >. } )
24	9, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23	bnj1467 32068	. . . . . . . . . . . 12 |- ( w e. Q -> A. d w e. Q )
25	24	nfcii 2579	. . . . . . . . . . 11 |- F/_ d Q
26		nfcv 2589	. . . . . . . . . . 11 |- F/_ d E
27	25, 26	nffn 5526	. . . . . . . . . 10 |- F/ d Q Fn E
28		bnj1463.13	. . . . . . . . . . . . 13 |- W = <. z , ( Q |` pred ( z , A , R ) ) >.
29	9, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 28	bnj1446 32059	. . . . . . . . . . . 12 |- ( ( Q ` z ) = ( G ` W ) -> A. d ( Q ` z ) = ( G ` W ) )
30	29	nfi 1596	. . . . . . . . . . 11 |- F/ d ( Q ` z ) = ( G ` W )
31	26, 30	nfral 2788	. . . . . . . . . 10 |- F/ d A. z e. E ( Q ` z ) = ( G ` W )
32	27, 31	nfan 1861	. . . . . . . . 9 |- F/ d ( Q Fn E /\ A. z e. E ( Q ` z ) = ( G ` W ) )
33	12, 32	nfan 1861	. . . . . . . 8 |- F/ d ( E e. B /\ ( Q Fn E /\ A. z e. E ( Q ` z ) = ( G ` W ) ) )
34	33	nfri 1808	. . . . . . 7 |- ( ( E e. B /\ ( Q Fn E /\ A. z e. E ( Q ` z ) = ( G ` W ) ) ) -> A. d ( E e. B /\ ( Q Fn E /\ A. z e. E ( Q ` z ) = ( G ` W ) ) ) )
35		bnj1463.17	. . . . . . . 8 |- ( ch -> Q Fn E )
36		bnj1463.16	. . . . . . . 8 |- ( ch -> A. z e. E ( Q ` z ) = ( G ` W ) )
37	1, 35, 36	jca32 535	. . . . . . 7 |- ( ch -> ( E e. B /\ ( Q Fn E /\ A. z e. E ( Q ` z ) = ( G ` W ) ) ) )
38	8, 34, 37	bnj1465 31861	. . . . . 6 |- ( ( ch /\ E e. _V ) -> E. d ( d e. B /\ ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) ) ) )
39	3, 38	mpdan 668	. . . . 5 |- ( ch -> E. d ( d e. B /\ ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) ) ) )
40		df-rex 2740	. . . . 5 |- ( E. d e. B ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) ) <-> E. d ( d e. B /\ ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) ) ) )
41	39, 40	sylibr 212	. . . 4 |- ( ch -> E. d e. B ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) ) )
42		bnj1463.15	. . . . 5 |- ( ch -> Q e. _V )
43		nfcv 2589	. . . . . . . 8 |- F/_ f B
44	9, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23	bnj1466 32067	. . . . . . . . . . 11 |- ( w e. Q -> A. f w e. Q )
45	44	nfcii 2579	. . . . . . . . . 10 |- F/_ f Q
46		nfcv 2589	. . . . . . . . . 10 |- F/_ f d
47	45, 46	nffn 5526	. . . . . . . . 9 |- F/ f Q Fn d
48	9, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 28	bnj1448 32061	. . . . . . . . . . 11 |- ( ( Q ` z ) = ( G ` W ) -> A. f ( Q ` z ) = ( G ` W ) )
49	48	nfi 1596	. . . . . . . . . 10 |- F/ f ( Q ` z ) = ( G ` W )
50	46, 49	nfral 2788	. . . . . . . . 9 |- F/ f A. z e. d ( Q ` z ) = ( G ` W )
51	47, 50	nfan 1861	. . . . . . . 8 |- F/ f ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) )
52	43, 51	nfrex 2790	. . . . . . 7 |- F/ f E. d e. B ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) )
53	52	nfri 1808	. . . . . 6 |- ( E. d e. B ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) ) -> A. f E. d e. B ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) ) )
54	25	nfeq2 2605	. . . . . . 7 |- F/ d f = Q
55		fneq1 5518	. . . . . . . 8 |- ( f = Q -> ( f Fn d <-> Q Fn d ) )
56		fveq1 5709	. . . . . . . . . 10 |- ( f = Q -> ( f ` z ) = ( Q ` z ) )
57		reseq1 5123	. . . . . . . . . . . . 13 |- ( f = Q -> ( f |` pred ( z , A , R ) ) = ( Q |` pred ( z , A , R ) ) )
58	57	opeq2d 4085	. . . . . . . . . . . 12 |- ( f = Q -> <. z , ( f |` pred ( z , A , R ) ) >. = <. z , ( Q |` pred ( z , A , R ) ) >. )
59	58, 28	syl6eqr 2493	. . . . . . . . . . 11 |- ( f = Q -> <. z , ( f |` pred ( z , A , R ) ) >. = W )
60	59	fveq2d 5714	. . . . . . . . . 10 |- ( f = Q -> ( G ` <. z , ( f |` pred ( z , A , R ) ) >. ) = ( G ` W ) )
61	56, 60	eqeq12d 2457	. . . . . . . . 9 |- ( f = Q -> ( ( f ` z ) = ( G ` <. z , ( f |` pred ( z , A , R ) ) >. ) <-> ( Q ` z ) = ( G ` W ) ) )
62	61	ralbidv 2754	. . . . . . . 8 |- ( f = Q -> ( A. z e. d ( f ` z ) = ( G ` <. z , ( f |` pred ( z , A , R ) ) >. ) <-> A. z e. d ( Q ` z ) = ( G ` W ) ) )
63	55, 62	anbi12d 710	. . . . . . 7 |- ( f = Q -> ( ( f Fn d /\ A. z e. d ( f ` z ) = ( G ` <. z , ( f |` pred ( z , A , R ) ) >. ) ) <-> ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) ) ) )
64	54, 63	rexbid 2753	. . . . . 6 |- ( f = Q -> ( E. d e. B ( f Fn d /\ A. z e. d ( f ` z ) = ( G ` <. z , ( f |` pred ( z , A , R ) ) >. ) ) <-> E. d e. B ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) ) ) )
65	53, 64, 44	bnj1468 31862	. . . . 5 |- ( Q e. _V -> ( [. Q / f ]. E. d e. B ( f Fn d /\ A. z e. d ( f ` z ) = ( G ` <. z , ( f |` pred ( z , A , R ) ) >. ) ) <-> E. d e. B ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) ) ) )
66	42, 65	syl 16	. . . 4 |- ( ch -> ( [. Q / f ]. E. d e. B ( f Fn d /\ A. z e. d ( f ` z ) = ( G ` <. z , ( f |` pred ( z , A , R ) ) >. ) ) <-> E. d e. B ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) ) ) )
67	41, 66	mpbird 232	. . 3 |- ( ch -> [. Q / f ]. E. d e. B ( f Fn d /\ A. z e. d ( f ` z ) = ( G ` <. z , ( f |` pred ( z , A , R ) ) >. ) ) )
68		fveq2 5710	. . . . . . . 8 |- ( x = z -> ( f ` x ) = ( f ` z ) )
69		id 22	. . . . . . . . . . 11 |- ( x = z -> x = z )
70		bnj602 31931	. . . . . . . . . . . 12 |- ( x = z -> pred ( x , A , R ) = pred ( z , A , R ) )
71	70	reseq2d 5129	. . . . . . . . . . 11 |- ( x = z -> ( f |` pred ( x , A , R ) ) = ( f |` pred ( z , A , R ) ) )
72	69, 71	opeq12d 4086	. . . . . . . . . 10 |- ( x = z -> <. x , ( f |` pred ( x , A , R ) ) >. = <. z , ( f |` pred ( z , A , R ) ) >. )
73	13, 72	syl5eq 2487	. . . . . . . . 9 |- ( x = z -> Y = <. z , ( f |` pred ( z , A , R ) ) >. )
74	73	fveq2d 5714	. . . . . . . 8 |- ( x = z -> ( G ` Y ) = ( G ` <. z , ( f |` pred ( z , A , R ) ) >. ) )
75	68, 74	eqeq12d 2457	. . . . . . 7 |- ( x = z -> ( ( f ` x ) = ( G ` Y ) <-> ( f ` z ) = ( G ` <. z , ( f |` pred ( z , A , R ) ) >. ) ) )
76	75	cbvralv 2966	. . . . . 6 |- ( A. x e. d ( f ` x ) = ( G ` Y ) <-> A. z e. d ( f ` z ) = ( G ` <. z , ( f |` pred ( z , A , R ) ) >. ) )
77	76	anbi2i 694	. . . . 5 |- ( ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) <-> ( f Fn d /\ A. z e. d ( f ` z ) = ( G ` <. z , ( f |` pred ( z , A , R ) ) >. ) ) )
78	77	rexbii 2759	. . . 4 |- ( E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) <-> E. d e. B ( f Fn d /\ A. z e. d ( f ` z ) = ( G ` <. z , ( f |` pred ( z , A , R ) ) >. ) ) )
79	78	sbcbii 3265	. . 3 |- ( [. Q / f ]. E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) <-> [. Q / f ]. E. d e. B ( f Fn d /\ A. z e. d ( f ` z ) = ( G ` <. z , ( f |` pred ( z , A , R ) ) >. ) ) )
80	67, 79	sylibr 212	. 2 |- ( ch -> [. Q / f ]. E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) )
81	14	bnj1454 31858	. . 3 |- ( Q e. _V -> ( Q e. C <-> [. Q / f ]. E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) ) )
82	42, 81	syl 16	. 2 |- ( ch -> ( Q e. C <-> [. Q / f ]. E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) ) )
83	80, 82	mpbird 232	1 |- ( ch -> Q e. C )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1369   E.wex 1586   e. wcel 1756   {cab 2429   =/= wne 2620   A.wral 2734   E.wrex 2735   {crab 2738   _Vcvv 2991   [.wsbc 3205   u. cun 3345   C_ wss 3347   (/)c0 3656   {csn 3896   <.cop 3902   U.cuni 4110   class class class wbr 4311   dom cdm 4859   |` cres 4861   Fn wfn 5432   ` cfv 5437   predc-bnj14 31699   FrSe w-bnj15 31703   trClc-bnj18 31705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ral 2739  df-rex 2740  df-rab 2743  df-v 2993  df-sbc 3206  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-br 4312  df-opab 4370  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-res 4871  df-iota 5400  df-fun 5439  df-fn 5440  df-fv 5445  df-bnj14 31700

This theorem is referenced by:  bnj1312  32072