Theorem bnj1463 31123

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
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	1	elexd 3214	. . . . . 6 |- ( ch -> E e. _V )
3		eleq1 2689	. . . . . . . 8 |- ( d = E -> ( d e. B <-> E e. B ) )
4		fneq2 5980	. . . . . . . . 9 |- ( d = E -> ( Q Fn d <-> Q Fn E ) )
5		raleq 3138	. . . . . . . . 9 |- ( d = E -> ( A. z e. d ( Q ` z ) = ( G ` W ) <-> A. z e. E ( Q ` z ) = ( G ` W ) ) )
6	4, 5	anbi12d 747	. . . . . . . 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 ) ) ) )
7	3, 6	anbi12d 747	. . . . . . 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 ) ) ) ) )
8		bnj1463.1	. . . . . . . . . . . 12 |- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
9	8	bnj1317 30892	. . . . . . . . . . 11 |- ( w e. B -> A. d w e. B )
10	9	nfcii 2755	. . . . . . . . . 10 |- F/_ d B
11	10	nfel2 2781	. . . . . . . . 9 |- F/ d E e. B
12		bnj1463.2	. . . . . . . . . . . . 13 |- Y = <. x , ( f |` pred ( x , A , R ) ) >.
13		bnj1463.3	. . . . . . . . . . . . 13 |- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
14		bnj1463.4	. . . . . . . . . . . . 13 |- ( ta <-> ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
15		bnj1463.5	. . . . . . . . . . . . 13 |- D = { x e. A | -. E. f ta }
16		bnj1463.6	. . . . . . . . . . . . 13 |- ( ps <-> ( R FrSe A /\ D =/= (/) ) )
17		bnj1463.7	. . . . . . . . . . . . 13 |- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
18		bnj1463.8	. . . . . . . . . . . . 13 |- ( ta' <-> [. y / x ]. ta )
19		bnj1463.9	. . . . . . . . . . . . 13 |- H = { f | E. y e. pred ( x , A , R ) ta' }
20		bnj1463.10	. . . . . . . . . . . . 13 |- P = U. H
21		bnj1463.11	. . . . . . . . . . . . 13 |- Z = <. x , ( P |` pred ( x , A , R ) ) >.
22		bnj1463.12	. . . . . . . . . . . . 13 |- Q = ( P u. { <. x , ( G ` Z ) >. } )
23	8, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22	bnj1467 31122	. . . . . . . . . . . 12 |- ( w e. Q -> A. d w e. Q )
24	23	nfcii 2755	. . . . . . . . . . 11 |- F/_ d Q
25		nfcv 2764	. . . . . . . . . . 11 |- F/_ d E
26	24, 25	nffn 5987	. . . . . . . . . 10 |- F/ d Q Fn E
27		bnj1463.13	. . . . . . . . . . . . 13 |- W = <. z , ( Q |` pred ( z , A , R ) ) >.
28	8, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 27	bnj1446 31113	. . . . . . . . . . . 12 |- ( ( Q ` z ) = ( G ` W ) -> A. d ( Q ` z ) = ( G ` W ) )
29	28	nf5i 2024	. . . . . . . . . . 11 |- F/ d ( Q ` z ) = ( G ` W )
30	25, 29	nfral 2945	. . . . . . . . . 10 |- F/ d A. z e. E ( Q ` z ) = ( G ` W )
31	26, 30	nfan 1828	. . . . . . . . 9 |- F/ d ( Q Fn E /\ A. z e. E ( Q ` z ) = ( G ` W ) )
32	11, 31	nfan 1828	. . . . . . . 8 |- F/ d ( E e. B /\ ( Q Fn E /\ A. z e. E ( Q ` z ) = ( G ` W ) ) )
33	32	nf5ri 2065	. . . . . . 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 ) ) ) )
34		bnj1463.17	. . . . . . . 8 |- ( ch -> Q Fn E )
35		bnj1463.16	. . . . . . . 8 |- ( ch -> A. z e. E ( Q ` z ) = ( G ` W ) )
36	1, 34, 35	jca32 558	. . . . . . 7 |- ( ch -> ( E e. B /\ ( Q Fn E /\ A. z e. E ( Q ` z ) = ( G ` W ) ) ) )
37	7, 33, 36	bnj1465 30915	. . . . . 6 |- ( ( ch /\ E e. _V ) -> E. d ( d e. B /\ ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) ) ) )
38	2, 37	mpdan 702	. . . . 5 |- ( ch -> E. d ( d e. B /\ ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) ) ) )
39		df-rex 2918	. . . . 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 ) ) ) )
40	38, 39	sylibr 224	. . . 4 |- ( ch -> E. d e. B ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) ) )
41		bnj1463.15	. . . . 5 |- ( ch -> Q e. _V )
42		nfcv 2764	. . . . . . . 8 |- F/_ f B
43	8, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22	bnj1466 31121	. . . . . . . . . . 11 |- ( w e. Q -> A. f w e. Q )
44	43	nfcii 2755	. . . . . . . . . 10 |- F/_ f Q
45		nfcv 2764	. . . . . . . . . 10 |- F/_ f d
46	44, 45	nffn 5987	. . . . . . . . 9 |- F/ f Q Fn d
47	8, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 27	bnj1448 31115	. . . . . . . . . . 11 |- ( ( Q ` z ) = ( G ` W ) -> A. f ( Q ` z ) = ( G ` W ) )
48	47	nf5i 2024	. . . . . . . . . 10 |- F/ f ( Q ` z ) = ( G ` W )
49	45, 48	nfral 2945	. . . . . . . . 9 |- F/ f A. z e. d ( Q ` z ) = ( G ` W )
50	46, 49	nfan 1828	. . . . . . . 8 |- F/ f ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) )
51	42, 50	nfrex 3007	. . . . . . 7 |- F/ f E. d e. B ( Q Fn d /\ A. z e. d ( Q ` z ) = ( G ` W ) )
52	51	nf5ri 2065	. . . . . 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 ) ) )
53	24	nfeq2 2780	. . . . . . 7 |- F/ d f = Q
54		fneq1 5979	. . . . . . . 8 |- ( f = Q -> ( f Fn d <-> Q Fn d ) )
55		fveq1 6190	. . . . . . . . . 10 |- ( f = Q -> ( f ` z ) = ( Q ` z ) )
56		reseq1 5390	. . . . . . . . . . . . 13 |- ( f = Q -> ( f |` pred ( z , A , R ) ) = ( Q |` pred ( z , A , R ) ) )
57	56	opeq2d 4409	. . . . . . . . . . . 12 |- ( f = Q -> <. z , ( f |` pred ( z , A , R ) ) >. = <. z , ( Q |` pred ( z , A , R ) ) >. )
58	57, 27	syl6eqr 2674	. . . . . . . . . . 11 |- ( f = Q -> <. z , ( f |` pred ( z , A , R ) ) >. = W )
59	58	fveq2d 6195	. . . . . . . . . 10 |- ( f = Q -> ( G ` <. z , ( f |` pred ( z , A , R ) ) >. ) = ( G ` W ) )
60	55, 59	eqeq12d 2637	. . . . . . . . 9 |- ( f = Q -> ( ( f ` z ) = ( G ` <. z , ( f |` pred ( z , A , R ) ) >. ) <-> ( Q ` z ) = ( G ` W ) ) )
61	60	ralbidv 2986	. . . . . . . 8 |- ( f = Q -> ( A. z e. d ( f ` z ) = ( G ` <. z , ( f |` pred ( z , A , R ) ) >. ) <-> A. z e. d ( Q ` z ) = ( G ` W ) ) )
62	54, 61	anbi12d 747	. . . . . . 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 ) ) ) )
63	53, 62	rexbid 3051	. . . . . 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 ) ) ) )
64	52, 63, 43	bnj1468 30916	. . . . 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 ) ) ) )
65	41, 64	syl 17	. . . 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 ) ) ) )
66	40, 65	mpbird 247	. . 3 |- ( ch -> [. Q / f ]. E. d e. B ( f Fn d /\ A. z e. d ( f ` z ) = ( G ` <. z , ( f |` pred ( z , A , R ) ) >. ) ) )
67		fveq2 6191	. . . . . . . 8 |- ( x = z -> ( f ` x ) = ( f ` z ) )
68		id 22	. . . . . . . . . . 11 |- ( x = z -> x = z )
69		bnj602 30985	. . . . . . . . . . . 12 |- ( x = z -> pred ( x , A , R ) = pred ( z , A , R ) )
70	69	reseq2d 5396	. . . . . . . . . . 11 |- ( x = z -> ( f |` pred ( x , A , R ) ) = ( f |` pred ( z , A , R ) ) )
71	68, 70	opeq12d 4410	. . . . . . . . . 10 |- ( x = z -> <. x , ( f |` pred ( x , A , R ) ) >. = <. z , ( f |` pred ( z , A , R ) ) >. )
72	12, 71	syl5eq 2668	. . . . . . . . 9 |- ( x = z -> Y = <. z , ( f |` pred ( z , A , R ) ) >. )
73	72	fveq2d 6195	. . . . . . . 8 |- ( x = z -> ( G ` Y ) = ( G ` <. z , ( f |` pred ( z , A , R ) ) >. ) )
74	67, 73	eqeq12d 2637	. . . . . . 7 |- ( x = z -> ( ( f ` x ) = ( G ` Y ) <-> ( f ` z ) = ( G ` <. z , ( f |` pred ( z , A , R ) ) >. ) ) )
75	74	cbvralv 3171	. . . . . 6 |- ( A. x e. d ( f ` x ) = ( G ` Y ) <-> A. z e. d ( f ` z ) = ( G ` <. z , ( f |` pred ( z , A , R ) ) >. ) )
76	75	anbi2i 730	. . . . 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 ) ) >. ) ) )
77	76	rexbii 3041	. . . 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 ) ) >. ) ) )
78	77	sbcbii 3491	. . 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 ) ) >. ) ) )
79	66, 78	sylibr 224	. 2 |- ( ch -> [. Q / f ]. E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) )
80	13	bnj1454 30912	. . 3 |- ( Q e. _V -> ( Q e. C <-> [. Q / f ]. E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) ) )
81	41, 80	syl 17	. 2 |- ( ch -> ( Q e. C <-> [. Q / f ]. E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) ) )
82	79, 81	mpbird 247	1 |- ( ch -> Q e. C )

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   _Vcvv 3200   [.wsbc 3435   u. cun 3572   C_ wss 3574   (/)c0 3915   {csn 4177   <.cop 4183   U.cuni 4436   class class class wbr 4653   dom cdm 5114   |` cres 5116   Fn wfn 5883   ` cfv 5888   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

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  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-br 4654  df-opab 4713  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-bnj14 30755

This theorem is referenced by:  bnj1312  31126