Theorem bnj849 30995

Description: Technical lemma for bnj69 31078. 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.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj849.1	|- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
bnj849.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj849.3	|- D = ( om \ { (/) } )
bnj849.4	|- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
bnj849.5	|- ( ch <-> ( R FrSe A /\ X e. A /\ n e. D ) )
bnj849.6	|- ( th <-> ( f Fn n /\ ph /\ ps ) )
bnj849.7	|- ( ph' <-> [. g / f ]. ph )
bnj849.8	|- ( ps' <-> [. g / f ]. ps )
bnj849.9	|- ( th' <-> [. g / f ]. th )
bnj849.10	|- ( ta <-> ( R FrSe A /\ X e. A ) )

Assertion

Ref	Expression
bnj849	|- ( ( R FrSe A /\ X e. A ) -> B e. _V )

Distinct variable groups:   A, f, i, n, y   B, g   D, f, g, n   D, i   R, f, i, n, y   f, X, n   ch, f, g   ph, g   ps, g   ta, g, n   th, g

Allowed substitution hints:   ph( y, f, i, n)   ps( y, f, i, n)   ch( y, i, n)   th( y, f, i, n)   ta( y, f, i)   A( g)   B( y, f, i, n)   D( y)   R( g)   X( y, g, i)   ph'( y, f, g, i, n)   ps'( y, f, g, i, n)   th'( y, f, g, i, n)

Proof of Theorem bnj849

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj849.10	. 2 |- ( ta <-> ( R FrSe A /\ X e. A ) )
2		bnj849.1	. . . 4 |- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
3		bnj849.2	. . . 4 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
4		bnj849.3	. . . 4 |- D = ( om \ { (/) } )
5		bnj849.5	. . . 4 |- ( ch <-> ( R FrSe A /\ X e. A /\ n e. D ) )
6		bnj849.6	. . . 4 |- ( th <-> ( f Fn n /\ ph /\ ps ) )
7	2, 3, 4, 5, 6	bnj865 30993	. . 3 |- E. w A. n ( ch -> E. f e. w th )
8		bnj849.4	. . . . . . . 8 |- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
9		bnj849.7	. . . . . . . 8 |- ( ph' <-> [. g / f ]. ph )
10		bnj849.8	. . . . . . . 8 |- ( ps' <-> [. g / f ]. ps )
11	8, 9, 10	bnj873 30994	. . . . . . 7 |- B = { g | E. n e. D ( g Fn n /\ ph' /\ ps' ) }
12		df-rex 2918	. . . . . . . . 9 |- ( E. n e. D ( g Fn n /\ ph' /\ ps' ) <-> E. n ( n e. D /\ ( g Fn n /\ ph' /\ ps' ) ) )
13		19.29 1801	. . . . . . . . . . 11 |- ( ( A. n ( ch -> E. f e. w th ) /\ E. n ( n e. D /\ ( g Fn n /\ ph' /\ ps' ) ) ) -> E. n ( ( ch -> E. f e. w th ) /\ ( n e. D /\ ( g Fn n /\ ph' /\ ps' ) ) ) )
14		an12 838	. . . . . . . . . . . . 13 |- ( ( ( ch -> E. f e. w th ) /\ ( n e. D /\ ( g Fn n /\ ph' /\ ps' ) ) ) <-> ( n e. D /\ ( ( ch -> E. f e. w th ) /\ ( g Fn n /\ ph' /\ ps' ) ) ) )
15		df-3an 1039	. . . . . . . . . . . . . . . 16 |- ( ( R FrSe A /\ X e. A /\ n e. D ) <-> ( ( R FrSe A /\ X e. A ) /\ n e. D ) )
16	1	anbi1i 731	. . . . . . . . . . . . . . . 16 |- ( ( ta /\ n e. D ) <-> ( ( R FrSe A /\ X e. A ) /\ n e. D ) )
17	15, 5, 16	3bitr4i 292	. . . . . . . . . . . . . . 15 |- ( ch <-> ( ta /\ n e. D ) )
18		id 22	. . . . . . . . . . . . . . . . 17 |- ( ch -> ch )
19		bnj849.9	. . . . . . . . . . . . . . . . . . . 20 |- ( th' <-> [. g / f ]. th )
20	6, 9, 10, 19	bnj581 30978	. . . . . . . . . . . . . . . . . . . 20 |- ( th' <-> ( g Fn n /\ ph' /\ ps' ) )
21	19, 20	bitr3i 266	. . . . . . . . . . . . . . . . . . 19 |- ( [. g / f ]. th <-> ( g Fn n /\ ph' /\ ps' ) )
22	2, 3, 4, 5, 6	bnj864 30992	. . . . . . . . . . . . . . . . . . . 20 |- ( ch -> E! f th )
23		df-rex 2918	. . . . . . . . . . . . . . . . . . . . 21 |- ( E. f e. w th <-> E. f ( f e. w /\ th ) )
24		exancom 1787	. . . . . . . . . . . . . . . . . . . . 21 |- ( E. f ( f e. w /\ th ) <-> E. f ( th /\ f e. w ) )
25	23, 24	sylbb 209	. . . . . . . . . . . . . . . . . . . 20 |- ( E. f e. w th -> E. f ( th /\ f e. w ) )
26		nfeu1 2480	. . . . . . . . . . . . . . . . . . . . . . 23 |- F/ f E! f th
27		nfe1 2027	. . . . . . . . . . . . . . . . . . . . . . 23 |- F/ f E. f ( th /\ f e. w )
28	26, 27	nfan 1828	. . . . . . . . . . . . . . . . . . . . . 22 |- F/ f ( E! f th /\ E. f ( th /\ f e. w ) )
29		nfsbc1v 3455	. . . . . . . . . . . . . . . . . . . . . . 23 |- F/ f [. g / f ]. th
30		nfv 1843	. . . . . . . . . . . . . . . . . . . . . . 23 |- F/ f g e. w
31	29, 30	nfim 1825	. . . . . . . . . . . . . . . . . . . . . 22 |- F/ f ( [. g / f ]. th -> g e. w )
32	28, 31	nfim 1825	. . . . . . . . . . . . . . . . . . . . 21 |- F/ f ( ( E! f th /\ E. f ( th /\ f e. w ) ) -> ( [. g / f ]. th -> g e. w ) )
33		sbceq1a 3446	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( f = g -> ( th <-> [. g / f ]. th ) )
34		elequ1 1997	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( f = g -> ( f e. w <-> g e. w ) )
35	33, 34	imbi12d 334	. . . . . . . . . . . . . . . . . . . . . 22 |- ( f = g -> ( ( th -> f e. w ) <-> ( [. g / f ]. th -> g e. w ) ) )
36	35	imbi2d 330	. . . . . . . . . . . . . . . . . . . . 21 |- ( f = g -> ( ( ( E! f th /\ E. f ( th /\ f e. w ) ) -> ( th -> f e. w ) ) <-> ( ( E! f th /\ E. f ( th /\ f e. w ) ) -> ( [. g / f ]. th -> g e. w ) ) ) )
37		eupick 2536	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( E! f th /\ E. f ( th /\ f e. w ) ) -> ( th -> f e. w ) )
38	32, 36, 37	chvar 2262	. . . . . . . . . . . . . . . . . . . 20 |- ( ( E! f th /\ E. f ( th /\ f e. w ) ) -> ( [. g / f ]. th -> g e. w ) )
39	22, 25, 38	syl2an 494	. . . . . . . . . . . . . . . . . . 19 |- ( ( ch /\ E. f e. w th ) -> ( [. g / f ]. th -> g e. w ) )
40	21, 39	syl5bir 233	. . . . . . . . . . . . . . . . . 18 |- ( ( ch /\ E. f e. w th ) -> ( ( g Fn n /\ ph' /\ ps' ) -> g e. w ) )
41	40	ex 450	. . . . . . . . . . . . . . . . 17 |- ( ch -> ( E. f e. w th -> ( ( g Fn n /\ ph' /\ ps' ) -> g e. w ) ) )
42	18, 41	embantd 59	. . . . . . . . . . . . . . . 16 |- ( ch -> ( ( ch -> E. f e. w th ) -> ( ( g Fn n /\ ph' /\ ps' ) -> g e. w ) ) )
43	42	impd 447	. . . . . . . . . . . . . . 15 |- ( ch -> ( ( ( ch -> E. f e. w th ) /\ ( g Fn n /\ ph' /\ ps' ) ) -> g e. w ) )
44	17, 43	sylbir 225	. . . . . . . . . . . . . 14 |- ( ( ta /\ n e. D ) -> ( ( ( ch -> E. f e. w th ) /\ ( g Fn n /\ ph' /\ ps' ) ) -> g e. w ) )
45	44	expimpd 629	. . . . . . . . . . . . 13 |- ( ta -> ( ( n e. D /\ ( ( ch -> E. f e. w th ) /\ ( g Fn n /\ ph' /\ ps' ) ) ) -> g e. w ) )
46	14, 45	syl5bi 232	. . . . . . . . . . . 12 |- ( ta -> ( ( ( ch -> E. f e. w th ) /\ ( n e. D /\ ( g Fn n /\ ph' /\ ps' ) ) ) -> g e. w ) )
47	46	exlimdv 1861	. . . . . . . . . . 11 |- ( ta -> ( E. n ( ( ch -> E. f e. w th ) /\ ( n e. D /\ ( g Fn n /\ ph' /\ ps' ) ) ) -> g e. w ) )
48	13, 47	syl5 34	. . . . . . . . . 10 |- ( ta -> ( ( A. n ( ch -> E. f e. w th ) /\ E. n ( n e. D /\ ( g Fn n /\ ph' /\ ps' ) ) ) -> g e. w ) )
49	48	expdimp 453	. . . . . . . . 9 |- ( ( ta /\ A. n ( ch -> E. f e. w th ) ) -> ( E. n ( n e. D /\ ( g Fn n /\ ph' /\ ps' ) ) -> g e. w ) )
50	12, 49	syl5bi 232	. . . . . . . 8 |- ( ( ta /\ A. n ( ch -> E. f e. w th ) ) -> ( E. n e. D ( g Fn n /\ ph' /\ ps' ) -> g e. w ) )
51	50	abssdv 3676	. . . . . . 7 |- ( ( ta /\ A. n ( ch -> E. f e. w th ) ) -> { g | E. n e. D ( g Fn n /\ ph' /\ ps' ) } C_ w )
52	11, 51	syl5eqss 3649	. . . . . 6 |- ( ( ta /\ A. n ( ch -> E. f e. w th ) ) -> B C_ w )
53		vex 3203	. . . . . . 7 |- w e. _V
54	53	ssex 4802	. . . . . 6 |- ( B C_ w -> B e. _V )
55	52, 54	syl 17	. . . . 5 |- ( ( ta /\ A. n ( ch -> E. f e. w th ) ) -> B e. _V )
56	55	ex 450	. . . 4 |- ( ta -> ( A. n ( ch -> E. f e. w th ) -> B e. _V ) )
57	56	exlimdv 1861	. . 3 |- ( ta -> ( E. w A. n ( ch -> E. f e. w th ) -> B e. _V ) )
58	7, 57	mpi 20	. 2 |- ( ta -> B e. _V )
59	1, 58	sylbir 225	1 |- ( ( R FrSe A /\ X e. A ) -> B e. _V )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   E!weu 2470   {cab 2608   A.wral 2912   E.wrex 2913   _Vcvv 3200   [.wsbc 3435   \ cdif 3571   C_ wss 3574   (/)c0 3915   {csn 4177   U_ciun 4520   suc csuc 5725   Fn wfn 5883   ` cfv 5888   omcom 7065   predc-bnj14 30754   FrSe w-bnj15 30758

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

This theorem is referenced by:  bnj893  30998