Theorem bnj849 34105

Description: Technical lemma for bnj69 34188. 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 34103	. . 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 34104	. . . . . . 7 |- B = { g | E. n e. D ( g Fn n /\ ph' /\ ps' ) }
12		df-rex 2813	. . . . . . . . 9 |- ( E. n e. D ( g Fn n /\ ph' /\ ps' ) <-> E. n ( n e. D /\ ( g Fn n /\ ph' /\ ps' ) ) )
13		19.29 1684	. . . . . . . . . . 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 797	. . . . . . . . . . . . 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 34088	. . . . . . . . . . . . . . . . . . . 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 34102	. . . . . . . . . . . . . . . . . . . 20 |- ( ch -> E! f th )
23		df-rex 2813	. . . . . . . . . . . . . . . . . . . . 21 |- ( E. f e. w th <-> E. f ( f e. w /\ th ) )
24		exancom 1672	. . . . . . . . . . . . . . . . . . . . 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 2295	. . . . . . . . . . . . . . . . . . . . . . 23 |- F/ f E! f th
27		nfe1 1841	. . . . . . . . . . . . . . . . . . . . . . 23 |- F/ f E. f ( th /\ f e. w )
28	26, 27	nfan 1929	. . . . . . . . . . . . . . . . . . . . . 22 |- F/ f ( E! f th /\ E. f ( th /\ f e. w ) )
29		nfsbc1v 3347	. . . . . . . . . . . . . . . . . . . . . . 23 |- F/ f [. g / f ]. th
30		nfv 1708	. . . . . . . . . . . . . . . . . . . . . . 23 |- F/ f g e. w
31	29, 30	nfim 1921	. . . . . . . . . . . . . . . . . . . . . 22 |- F/ f ( [. g / f ]. th -> g e. w )
32	28, 31	nfim 1921	. . . . . . . . . . . . . . . . . . . . 21 |- F/ f ( ( E! f th /\ E. f ( th /\ f e. w ) ) -> ( [. g / f ]. th -> g e. w ) )
33		sbceq1a 3338	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( f = g -> ( th <-> [. g / f ]. th ) )
34		elequ1 1822	. . . . . . . . . . . . . . . . . . . . . . 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 2358	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( E! f th /\ E. f ( th /\ f e. w ) ) -> ( th -> f e. w ) )
38	32, 36, 37	chvar 2014	. . . . . . . . . . . . . . . . . . . 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 1725	. . . . . . . . . . 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 3570	. . . . . . 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 3543	. . . . . 6 |- ( ( ta /\ A. n ( ch -> E. f e. w th ) ) -> B C_ w )
53		vex 3112	. . . . . . 7 |- w e. _V
54	53	ssex 4600	. . . . . 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 1725	. . 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 1393   = wceq 1395   E.wex 1613   e. wcel 1819   E!weu 2283   {cab 2442   A.wral 2807   E.wrex 2808   _Vcvv 3109   [.wsbc 3327   \ cdif 3468   C_ wss 3471   (/)c0 3793   {csn 4032   U_ciun 4332   suc csuc 4889   Fn wfn 5589   ` cfv 5594   omcom 6699   predc-bnj14 33862   FrSe w-bnj15 33866

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-reg 8036  ax-inf2 8075

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6700  df-1o 7148  df-bnj17 33861  df-bnj14 33863  df-bnj13 33865  df-bnj15 33867

This theorem is referenced by:  bnj893  34108