Theorem bnj1489 33209

Description: Technical lemma for bnj60 33215. 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
bnj1489.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj1489.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj1489.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1489.4	|- ( ta <-> ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
bnj1489.5	|- D = { x e. A | -. E. f ta }
bnj1489.6	|- ( ps <-> ( R FrSe A /\ D =/= (/) ) )
bnj1489.7	|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
bnj1489.8	|- ( ta' <-> [. y / x ]. ta )
bnj1489.9	|- H = { f | E. y e. pred ( x , A , R ) ta' }
bnj1489.10	|- P = U. H
bnj1489.11	|- Z = <. x , ( P |` pred ( x , A , R ) ) >.
bnj1489.12	|- Q = ( P u. { <. x , ( G ` Z ) >. } )

Assertion

Ref	Expression
bnj1489	|- ( ch -> Q e. _V )

Distinct variable groups:   A, d, f, x   y, A, f, x   B, f   y, D   G, d, f   R, d, f, x   y, R   ps, y   ta, y

Allowed substitution hints:   ps( x, f, d)   ch( x, y, f, d)   ta( x, f, d)   B( x, y, d)   C( x, y, f, d)   D( x, f, d)   P( x, y, f, d)   Q( x, y, f, d)   G( x, y)   H( x, y, f, d)   Y( x, y, f, d)   Z( x, y, f, d)   ta'( x, y, f, d)

Proof of Theorem bnj1489

Step	Hyp	Ref	Expression
1		bnj1489.12	. 2 |- Q = ( P u. { <. x , ( G ` Z ) >. } )
2		bnj1489.10	. . . 4 |- P = U. H
3		bnj1489.7	. . . . . . . 8 |- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
4		bnj1489.6	. . . . . . . . 9 |- ( ps <-> ( R FrSe A /\ D =/= (/) ) )
5		bnj1364 33181	. . . . . . . . . 10 |- ( R FrSe A -> R Se A )
6		df-bnj13 32841	. . . . . . . . . 10 |- ( R Se A <-> A. x e. A pred ( x , A , R ) e. _V )
7	5, 6	sylib 196	. . . . . . . . 9 |- ( R FrSe A -> A. x e. A pred ( x , A , R ) e. _V )
8	4, 7	bnj832 32912	. . . . . . . 8 |- ( ps -> A. x e. A pred ( x , A , R ) e. _V )
9	3, 8	bnj835 32914	. . . . . . 7 |- ( ch -> A. x e. A pred ( x , A , R ) e. _V )
10		bnj1489.5	. . . . . . . 8 |- D = { x e. A | -. E. f ta }
11	10, 3	bnj1212 32955	. . . . . . 7 |- ( ch -> x e. A )
12	9, 11	bnj1294 32973	. . . . . 6 |- ( ch -> pred ( x , A , R ) e. _V )
13		nfv 1683	. . . . . . . . 9 |- F/ y ps
14		nfv 1683	. . . . . . . . 9 |- F/ y x e. D
15		nfra1 2845	. . . . . . . . 9 |- F/ y A. y e. D -. y R x
16	13, 14, 15	nf3an 1877	. . . . . . . 8 |- F/ y ( ps /\ x e. D /\ A. y e. D -. y R x )
17	3, 16	nfxfr 1625	. . . . . . 7 |- F/ y ch
18	4	simplbi 460	. . . . . . . . . . 11 |- ( ps -> R FrSe A )
19	3, 18	bnj835 32914	. . . . . . . . . 10 |- ( ch -> R FrSe A )
20	19	adantr 465	. . . . . . . . 9 |- ( ( ch /\ y e. pred ( x , A , R ) ) -> R FrSe A )
21		bnj1489.1	. . . . . . . . . . 11 |- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
22		bnj1489.2	. . . . . . . . . . 11 |- Y = <. x , ( f |` pred ( x , A , R ) ) >.
23		bnj1489.3	. . . . . . . . . . 11 |- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
24		bnj1489.4	. . . . . . . . . . 11 |- ( ta <-> ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
25		bnj1489.8	. . . . . . . . . . 11 |- ( ta' <-> [. y / x ]. ta )
26	21, 22, 23, 24, 10, 4, 3, 25	bnj1388 33186	. . . . . . . . . 10 |- ( ch -> A. y e. pred ( x , A , R ) E. f ta' )
27	26	r19.21bi 2833	. . . . . . . . 9 |- ( ( ch /\ y e. pred ( x , A , R ) ) -> E. f ta' )
28		nfv 1683	. . . . . . . . . . . 12 |- F/ x R FrSe A
29		nfsbc1v 3351	. . . . . . . . . . . . . 14 |- F/ x [. y / x ]. ta
30	25, 29	nfxfr 1625	. . . . . . . . . . . . 13 |- F/ x ta'
31	30	nfex 1895	. . . . . . . . . . . 12 |- F/ x E. f ta'
32	28, 31	nfan 1875	. . . . . . . . . . 11 |- F/ x ( R FrSe A /\ E. f ta' )
33	30	nfeu 2294	. . . . . . . . . . 11 |- F/ x E! f ta'
34	32, 33	nfim 1867	. . . . . . . . . 10 |- F/ x ( ( R FrSe A /\ E. f ta' ) -> E! f ta' )
35		sneq 4037	. . . . . . . . . . . . . . . . 17 |- ( x = y -> { x } = { y } )
36		bnj1318 33178	. . . . . . . . . . . . . . . . 17 |- ( x = y -> trCl ( x , A , R ) = trCl ( y , A , R ) )
37	35, 36	uneq12d 3659	. . . . . . . . . . . . . . . 16 |- ( x = y -> ( { x } u. trCl ( x , A , R ) ) = ( { y } u. trCl ( y , A , R ) ) )
38	37	eqeq2d 2481	. . . . . . . . . . . . . . 15 |- ( x = y -> ( dom f = ( { x } u. trCl ( x , A , R ) ) <-> dom f = ( { y } u. trCl ( y , A , R ) ) ) )
39	38	anbi2d 703	. . . . . . . . . . . . . 14 |- ( x = y -> ( ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) <-> ( f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) ) )
40	21, 22, 23, 24, 25	bnj1373 33183	. . . . . . . . . . . . . 14 |- ( ta' <-> ( f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) )
41	39, 40	syl6bbr 263	. . . . . . . . . . . . 13 |- ( x = y -> ( ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) <-> ta' ) )
42	41	exbidv 1690	. . . . . . . . . . . 12 |- ( x = y -> ( E. f ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) <-> E. f ta' ) )
43	42	anbi2d 703	. . . . . . . . . . 11 |- ( x = y -> ( ( R FrSe A /\ E. f ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) ) <-> ( R FrSe A /\ E. f ta' ) ) )
44	41	eubidv 2298	. . . . . . . . . . 11 |- ( x = y -> ( E! f ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) <-> E! f ta' ) )
45	43, 44	imbi12d 320	. . . . . . . . . 10 |- ( x = y -> ( ( ( R FrSe A /\ E. f ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) ) -> E! f ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) ) <-> ( ( R FrSe A /\ E. f ta' ) -> E! f ta' ) ) )
46		biid 236	. . . . . . . . . . 11 |- ( ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) <-> ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
47	21, 22, 23, 46	bnj1321 33180	. . . . . . . . . 10 |- ( ( R FrSe A /\ E. f ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) ) -> E! f ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
48	34, 45, 47	chvar 1982	. . . . . . . . 9 |- ( ( R FrSe A /\ E. f ta' ) -> E! f ta' )
49	20, 27, 48	syl2anc 661	. . . . . . . 8 |- ( ( ch /\ y e. pred ( x , A , R ) ) -> E! f ta' )
50	49	ex 434	. . . . . . 7 |- ( ch -> ( y e. pred ( x , A , R ) -> E! f ta' ) )
51	17, 50	ralrimi 2864	. . . . . 6 |- ( ch -> A. y e. pred ( x , A , R ) E! f ta' )
52		bnj1489.9	. . . . . . 7 |- H = { f | E. y e. pred ( x , A , R ) ta' }
53	52	a1i 11	. . . . . 6 |- ( ch -> H = { f | E. y e. pred ( x , A , R ) ta' } )
54		biid 236	. . . . . . 7 |- ( ( pred ( x , A , R ) e. _V /\ A. y e. pred ( x , A , R ) E! f ta' /\ H = { f | E. y e. pred ( x , A , R ) ta' } ) <-> ( pred ( x , A , R ) e. _V /\ A. y e. pred ( x , A , R ) E! f ta' /\ H = { f | E. y e. pred ( x , A , R ) ta' } ) )
55	54	bnj1366 32985	. . . . . 6 |- ( ( pred ( x , A , R ) e. _V /\ A. y e. pred ( x , A , R ) E! f ta' /\ H = { f | E. y e. pred ( x , A , R ) ta' } ) -> H e. _V )
56	12, 51, 53, 55	syl3anc 1228	. . . . 5 |- ( ch -> H e. _V )
57		uniexg 6581	. . . . 5 |- ( H e. _V -> U. H e. _V )
58	56, 57	syl 16	. . . 4 |- ( ch -> U. H e. _V )
59	2, 58	syl5eqel 2559	. . 3 |- ( ch -> P e. _V )
60		snex 4688	. . . 4 |- { <. x , ( G ` Z ) >. } e. _V
61	60	a1i 11	. . 3 |- ( ch -> { <. x , ( G ` Z ) >. } e. _V )
62	59, 61	bnj1149 32948	. 2 |- ( ch -> ( P u. { <. x , ( G ` Z ) >. } ) e. _V )
63	1, 62	syl5eqel 2559	1 |- ( ch -> Q e. _V )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   E.wex 1596   e. wcel 1767   E!weu 2275   {cab 2452   =/= wne 2662   A.wral 2814   E.wrex 2815   {crab 2818   _Vcvv 3113   [.wsbc 3331   u. cun 3474   C_ wss 3476   (/)c0 3785   {csn 4027   <.cop 4033   U.cuni 4245   class class class wbr 4447   dom cdm 4999   |` cres 5001   Fn wfn 5583   ` cfv 5588   predc-bnj14 32838   Se w-bnj13 32840   FrSe w-bnj15 32842   trClc-bnj18 32844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-reg 8018  ax-inf2 8058

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-om 6685  df-1o 7130  df-bnj17 32837  df-bnj14 32839  df-bnj13 32841  df-bnj15 32843  df-bnj18 32845  df-bnj19 32847

This theorem is referenced by:  bnj1312  33211