Theorem bnj849 33080

Description: Technical lemma for bnj69 33163. 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 33078	. . 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 33079	. . . . . . 7 |- B = { g | E. n e. D ( g Fn n /\ ph' /\ ps' ) }
12		df-rex 2820	. . . . . . . . 9 |- ( E. n e. D ( g Fn n /\ ph' /\ ps' ) <-> E. n ( n e. D /\ ( g Fn n /\ ph' /\ ps' ) ) )
13		19.29 1660	. . . . . . . . . . 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 795	. . . . . . . . . . . . 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 975	. . . . . . . . . . . . . . . 16 |- ( ( R FrSe A /\ X e. A /\ n e. D ) <-> ( ( R FrSe A /\ X e. A ) /\ n e. D ) )
16	1	anbi1i 695	. . . . . . . . . . . . . . . 16 |- ( ( ta /\ n e. D ) <-> ( ( R FrSe A /\ X e. A ) /\ n e. D ) )
17	15, 5, 16	3bitr4i 277	. . . . . . . . . . . . . . 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 33063	. . . . . . . . . . . . . . . . . . . 20 |- ( th' <-> ( g Fn n /\ ph' /\ ps' ) )
21	19, 20	bitr3i 251	. . . . . . . . . . . . . . . . . . 19 |- ( [. g / f ]. th <-> ( g Fn n /\ ph' /\ ps' ) )
22	2, 3, 4, 5, 6	bnj864 33077	. . . . . . . . . . . . . . . . . . . 20 |- ( ch -> E! f th )
23		df-rex 2820	. . . . . . . . . . . . . . . . . . . . 21 |- ( E. f e. w th <-> E. f ( f e. w /\ th ) )
24		exancom 1648	. . . . . . . . . . . . . . . . . . . . 21 |- ( E. f ( f e. w /\ th ) <-> E. f ( th /\ f e. w ) )
25	23, 24	sylbb 197	. . . . . . . . . . . . . . . . . . . 20 |- ( E. f e. w th -> E. f ( th /\ f e. w ) )
26		nfeu1 2288	. . . . . . . . . . . . . . . . . . . . . . 23 |- F/ f E! f th
27		nfe1 1789	. . . . . . . . . . . . . . . . . . . . . . 23 |- F/ f E. f ( th /\ f e. w )
28	26, 27	nfan 1875	. . . . . . . . . . . . . . . . . . . . . 22 |- F/ f ( E! f th /\ E. f ( th /\ f e. w ) )
29		nfsbc1v 3351	. . . . . . . . . . . . . . . . . . . . . . 23 |- F/ f [. g / f ]. th
30		nfv 1683	. . . . . . . . . . . . . . . . . . . . . . 23 |- F/ f g e. w
31	29, 30	nfim 1867	. . . . . . . . . . . . . . . . . . . . . 22 |- F/ f ( [. g / f ]. th -> g e. w )
32	28, 31	nfim 1867	. . . . . . . . . . . . . . . . . . . . 21 |- F/ f ( ( E! f th /\ E. f ( th /\ f e. w ) ) -> ( [. g / f ]. th -> g e. w ) )
33		sbceq1a 3342	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( f = g -> ( th <-> [. g / f ]. th ) )
34		elequ1 1770	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( f = g -> ( f e. w <-> g e. w ) )
35	33, 34	imbi12d 320	. . . . . . . . . . . . . . . . . . . . . 22 |- ( f = g -> ( ( th -> f e. w ) <-> ( [. g / f ]. th -> g e. w ) ) )
36	35	imbi2d 316	. . . . . . . . . . . . . . . . . . . . 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 2364	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( E! f th /\ E. f ( th /\ f e. w ) ) -> ( th -> f e. w ) )
38	32, 36, 37	chvar 1982	. . . . . . . . . . . . . . . . . . . 20 |- ( ( E! f th /\ E. f ( th /\ f e. w ) ) -> ( [. g / f ]. th -> g e. w ) )
39	22, 25, 38	syl2an 477	. . . . . . . . . . . . . . . . . . 19 |- ( ( ch /\ E. f e. w th ) -> ( [. g / f ]. th -> g e. w ) )
40	21, 39	syl5bir 218	. . . . . . . . . . . . . . . . . 18 |- ( ( ch /\ E. f e. w th ) -> ( ( g Fn n /\ ph' /\ ps' ) -> g e. w ) )
41	40	ex 434	. . . . . . . . . . . . . . . . 17 |- ( ch -> ( E. f e. w th -> ( ( g Fn n /\ ph' /\ ps' ) -> g e. w ) ) )
42	18, 41	embantd 54	. . . . . . . . . . . . . . . 16 |- ( ch -> ( ( ch -> E. f e. w th ) -> ( ( g Fn n /\ ph' /\ ps' ) -> g e. w ) ) )
43	42	impd 431	. . . . . . . . . . . . . . 15 |- ( ch -> ( ( ( ch -> E. f e. w th ) /\ ( g Fn n /\ ph' /\ ps' ) ) -> g e. w ) )
44	17, 43	sylbir 213	. . . . . . . . . . . . . 14 |- ( ( ta /\ n e. D ) -> ( ( ( ch -> E. f e. w th ) /\ ( g Fn n /\ ph' /\ ps' ) ) -> g e. w ) )
45	44	expimpd 603	. . . . . . . . . . . . 13 |- ( ta -> ( ( n e. D /\ ( ( ch -> E. f e. w th ) /\ ( g Fn n /\ ph' /\ ps' ) ) ) -> g e. w ) )
46	14, 45	syl5bi 217	. . . . . . . . . . . 12 |- ( ta -> ( ( ( ch -> E. f e. w th ) /\ ( n e. D /\ ( g Fn n /\ ph' /\ ps' ) ) ) -> g e. w ) )
47	46	exlimdv 1700	. . . . . . . . . . 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 32	. . . . . . . . . 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 437	. . . . . . . . 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 217	. . . . . . . 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 3574	. . . . . . 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 3548	. . . . . 6 |- ( ( ta /\ A. n ( ch -> E. f e. w th ) ) -> B C_ w )
53		vex 3116	. . . . . . 7 |- w e. _V
54	53	ssex 4591	. . . . . 6 |- ( B C_ w -> B e. _V )
55	52, 54	syl 16	. . . . 5 |- ( ( ta /\ A. n ( ch -> E. f e. w th ) ) -> B e. _V )
56	55	ex 434	. . . 4 |- ( ta -> ( A. n ( ch -> E. f e. w th ) -> B e. _V ) )
57	56	exlimdv 1700	. . 3 |- ( ta -> ( E. w A. n ( ch -> E. f e. w th ) -> B e. _V ) )
58	7, 57	mpi 17	. 2 |- ( ta -> B e. _V )
59	1, 58	sylbir 213	1 |- ( ( R FrSe A /\ X e. A ) -> B e. _V )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   A.wal 1377   = wceq 1379   E.wex 1596   e. wcel 1767   E!weu 2275   {cab 2452   A.wral 2814   E.wrex 2815   _Vcvv 3113   [.wsbc 3331   \ cdif 3473   C_ wss 3476   (/)c0 3785   {csn 4027   U_ciun 4325   suc csuc 4880   Fn wfn 5583   ` cfv 5588   omcom 6684   predc-bnj14 32838   FrSe w-bnj15 32842

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

This theorem is referenced by:  bnj893  33083