Theorem bnj1489 31124

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
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 31096	. . . . . . . . . 10 |- ( R FrSe A -> R Se A )
6		df-bnj13 30757	. . . . . . . . . 10 |- ( R Se A <-> A. x e. A pred ( x , A , R ) e. _V )
7	5, 6	sylib 208	. . . . . . . . 9 |- ( R FrSe A -> A. x e. A pred ( x , A , R ) e. _V )
8	4, 7	bnj832 30828	. . . . . . . 8 |- ( ps -> A. x e. A pred ( x , A , R ) e. _V )
9	3, 8	bnj835 30829	. . . . . . 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 30870	. . . . . . 7 |- ( ch -> x e. A )
12	9, 11	bnj1294 30888	. . . . . 6 |- ( ch -> pred ( x , A , R ) e. _V )
13		nfv 1843	. . . . . . . . 9 |- F/ y ps
14		nfv 1843	. . . . . . . . 9 |- F/ y x e. D
15		nfra1 2941	. . . . . . . . 9 |- F/ y A. y e. D -. y R x
16	13, 14, 15	nf3an 1831	. . . . . . . 8 |- F/ y ( ps /\ x e. D /\ A. y e. D -. y R x )
17	3, 16	nfxfr 1779	. . . . . . 7 |- F/ y ch
18	4	simplbi 476	. . . . . . . . . . 11 |- ( ps -> R FrSe A )
19	3, 18	bnj835 30829	. . . . . . . . . 10 |- ( ch -> R FrSe A )
20	19	adantr 481	. . . . . . . . 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 31101	. . . . . . . . . 10 |- ( ch -> A. y e. pred ( x , A , R ) E. f ta' )
27	26	r19.21bi 2932	. . . . . . . . 9 |- ( ( ch /\ y e. pred ( x , A , R ) ) -> E. f ta' )
28		nfv 1843	. . . . . . . . . . . 12 |- F/ x R FrSe A
29		nfsbc1v 3455	. . . . . . . . . . . . . 14 |- F/ x [. y / x ]. ta
30	25, 29	nfxfr 1779	. . . . . . . . . . . . 13 |- F/ x ta'
31	30	nfex 2154	. . . . . . . . . . . 12 |- F/ x E. f ta'
32	28, 31	nfan 1828	. . . . . . . . . . 11 |- F/ x ( R FrSe A /\ E. f ta' )
33	30	nfeu 2486	. . . . . . . . . . 11 |- F/ x E! f ta'
34	32, 33	nfim 1825	. . . . . . . . . 10 |- F/ x ( ( R FrSe A /\ E. f ta' ) -> E! f ta' )
35		sneq 4187	. . . . . . . . . . . . . . . . 17 |- ( x = y -> { x } = { y } )
36		bnj1318 31093	. . . . . . . . . . . . . . . . 17 |- ( x = y -> trCl ( x , A , R ) = trCl ( y , A , R ) )
37	35, 36	uneq12d 3768	. . . . . . . . . . . . . . . 16 |- ( x = y -> ( { x } u. trCl ( x , A , R ) ) = ( { y } u. trCl ( y , A , R ) ) )
38	37	eqeq2d 2632	. . . . . . . . . . . . . . 15 |- ( x = y -> ( dom f = ( { x } u. trCl ( x , A , R ) ) <-> dom f = ( { y } u. trCl ( y , A , R ) ) ) )
39	38	anbi2d 740	. . . . . . . . . . . . . 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 31098	. . . . . . . . . . . . . 14 |- ( ta' <-> ( f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) )
41	39, 40	syl6bbr 278	. . . . . . . . . . . . 13 |- ( x = y -> ( ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) <-> ta' ) )
42	41	exbidv 1850	. . . . . . . . . . . 12 |- ( x = y -> ( E. f ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) <-> E. f ta' ) )
43	42	anbi2d 740	. . . . . . . . . . 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 2490	. . . . . . . . . . 11 |- ( x = y -> ( E! f ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) <-> E! f ta' ) )
45	43, 44	imbi12d 334	. . . . . . . . . 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 251	. . . . . . . . . . 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 31095	. . . . . . . . . 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 2262	. . . . . . . . 9 |- ( ( R FrSe A /\ E. f ta' ) -> E! f ta' )
49	20, 27, 48	syl2anc 693	. . . . . . . 8 |- ( ( ch /\ y e. pred ( x , A , R ) ) -> E! f ta' )
50	49	ex 450	. . . . . . 7 |- ( ch -> ( y e. pred ( x , A , R ) -> E! f ta' ) )
51	17, 50	ralrimi 2957	. . . . . 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 251	. . . . . . 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 30900	. . . . . 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 1326	. . . . 5 |- ( ch -> H e. _V )
57		uniexg 6955	. . . . 5 |- ( H e. _V -> U. H e. _V )
58	56, 57	syl 17	. . . 4 |- ( ch -> U. H e. _V )
59	2, 58	syl5eqel 2705	. . 3 |- ( ch -> P e. _V )
60		snex 4908	. . . 4 |- { <. x , ( G ` Z ) >. } e. _V
61	60	a1i 11	. . 3 |- ( ch -> { <. x , ( G ` Z ) >. } e. _V )
62	59, 61	bnj1149 30863	. 2 |- ( ch -> ( P u. { <. x , ( G ` Z ) >. } ) e. _V )
63	1, 62	syl5eqel 2705	1 |- ( ch -> Q e. _V )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   E!weu 2470   {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   Se w-bnj13 30756   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-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:  bnj1312  31126