Theorem bnj1489 29867

Description: Technical lemma for bnj60 29873. 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 29839	. . . . . . . . . 10 |- ( R FrSe A -> R Se A )
6		df-bnj13 29498	. . . . . . . . . 10 |- ( R Se A <-> A. x e. A pred ( x , A , R ) e. _V )
7	5, 6	sylib 200	. . . . . . . . 9 |- ( R FrSe A -> A. x e. A pred ( x , A , R ) e. _V )
8	4, 7	bnj832 29570	. . . . . . . 8 |- ( ps -> A. x e. A pred ( x , A , R ) e. _V )
9	3, 8	bnj835 29572	. . . . . . 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 29613	. . . . . . 7 |- ( ch -> x e. A )
12	9, 11	bnj1294 29631	. . . . . 6 |- ( ch -> pred ( x , A , R ) e. _V )
13		nfv 1752	. . . . . . . . 9 |- F/ y ps
14		nfv 1752	. . . . . . . . 9 |- F/ y x e. D
15		nfra1 2807	. . . . . . . . 9 |- F/ y A. y e. D -. y R x
16	13, 14, 15	nf3an 1987	. . . . . . . 8 |- F/ y ( ps /\ x e. D /\ A. y e. D -. y R x )
17	3, 16	nfxfr 1693	. . . . . . 7 |- F/ y ch
18	4	simplbi 462	. . . . . . . . . . 11 |- ( ps -> R FrSe A )
19	3, 18	bnj835 29572	. . . . . . . . . 10 |- ( ch -> R FrSe A )
20	19	adantr 467	. . . . . . . . 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 29844	. . . . . . . . . 10 |- ( ch -> A. y e. pred ( x , A , R ) E. f ta' )
27	26	r19.21bi 2795	. . . . . . . . 9 |- ( ( ch /\ y e. pred ( x , A , R ) ) -> E. f ta' )
28		nfv 1752	. . . . . . . . . . . 12 |- F/ x R FrSe A
29		nfsbc1v 3320	. . . . . . . . . . . . . 14 |- F/ x [. y / x ]. ta
30	25, 29	nfxfr 1693	. . . . . . . . . . . . 13 |- F/ x ta'
31	30	nfex 2005	. . . . . . . . . . . 12 |- F/ x E. f ta'
32	28, 31	nfan 1985	. . . . . . . . . . 11 |- F/ x ( R FrSe A /\ E. f ta' )
33	30	nfeu 2283	. . . . . . . . . . 11 |- F/ x E! f ta'
34	32, 33	nfim 1977	. . . . . . . . . 10 |- F/ x ( ( R FrSe A /\ E. f ta' ) -> E! f ta' )
35		sneq 4007	. . . . . . . . . . . . . . . . 17 |- ( x = y -> { x } = { y } )
36		bnj1318 29836	. . . . . . . . . . . . . . . . 17 |- ( x = y -> trCl ( x , A , R ) = trCl ( y , A , R ) )
37	35, 36	uneq12d 3622	. . . . . . . . . . . . . . . 16 |- ( x = y -> ( { x } u. trCl ( x , A , R ) ) = ( { y } u. trCl ( y , A , R ) ) )
38	37	eqeq2d 2437	. . . . . . . . . . . . . . 15 |- ( x = y -> ( dom f = ( { x } u. trCl ( x , A , R ) ) <-> dom f = ( { y } u. trCl ( y , A , R ) ) ) )
39	38	anbi2d 709	. . . . . . . . . . . . . 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 29841	. . . . . . . . . . . . . 14 |- ( ta' <-> ( f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) )
41	39, 40	syl6bbr 267	. . . . . . . . . . . . 13 |- ( x = y -> ( ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) <-> ta' ) )
42	41	exbidv 1759	. . . . . . . . . . . 12 |- ( x = y -> ( E. f ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) <-> E. f ta' ) )
43	42	anbi2d 709	. . . . . . . . . . 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 2287	. . . . . . . . . . 11 |- ( x = y -> ( E! f ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) <-> E! f ta' ) )
45	43, 44	imbi12d 322	. . . . . . . . . 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 240	. . . . . . . . . . 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 29838	. . . . . . . . . 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 2068	. . . . . . . . 9 |- ( ( R FrSe A /\ E. f ta' ) -> E! f ta' )
49	20, 27, 48	syl2anc 666	. . . . . . . 8 |- ( ( ch /\ y e. pred ( x , A , R ) ) -> E! f ta' )
50	49	ex 436	. . . . . . 7 |- ( ch -> ( y e. pred ( x , A , R ) -> E! f ta' ) )
51	17, 50	ralrimi 2826	. . . . . 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 240	. . . . . . 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 29643	. . . . . 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 1265	. . . . 5 |- ( ch -> H e. _V )
57		uniexg 6600	. . . . 5 |- ( H e. _V -> U. H e. _V )
58	56, 57	syl 17	. . . 4 |- ( ch -> U. H e. _V )
59	2, 58	syl5eqel 2515	. . 3 |- ( ch -> P e. _V )
60		snex 4660	. . . 4 |- { <. x , ( G ` Z ) >. } e. _V
61	60	a1i 11	. . 3 |- ( ch -> { <. x , ( G ` Z ) >. } e. _V )
62	59, 61	bnj1149 29606	. 2 |- ( ch -> ( P u. { <. x , ( G ` Z ) >. } ) e. _V )
63	1, 62	syl5eqel 2515	1 |- ( ch -> Q e. _V )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 983   = wceq 1438   E.wex 1660   e. wcel 1869   E!weu 2266   {cab 2408   =/= wne 2619   A.wral 2776   E.wrex 2777   {crab 2780   _Vcvv 3082   [.wsbc 3300   u. cun 3435   C_ wss 3437   (/)c0 3762   {csn 3997   <.cop 4003   U.cuni 4217   class class class wbr 4421   dom cdm 4851   |` cres 4853   Fn wfn 5594   ` cfv 5599   predc-bnj14 29495   Se w-bnj13 29497   FrSe w-bnj15 29499   trClc-bnj18 29501

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-reg 8111  ax-inf2 8150

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-fal 1444  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-om 6705  df-1o 7188  df-bnj17 29494  df-bnj14 29496  df-bnj13 29498  df-bnj15 29500  df-bnj18 29502  df-bnj19 29504

This theorem is referenced by:  bnj1312  29869