Theorem bnj517 30955

Description: Technical lemma for bnj518 30956. 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
bnj517.1	|- ( ph <-> ( F ` (/) ) = pred ( X , A , R ) )
bnj517.2	|- ( ps <-> A. i e. om ( suc i e. N -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) )

Assertion

Ref	Expression
bnj517	|- ( ( N e. om /\ ph /\ ps ) -> A. n e. N ( F ` n ) C_ A )

Distinct variable groups:   i, n, y, A   i, F, n   i, N, n

Allowed substitution hints:   ph( y, i, n)   ps( y, i, n)   R( y, i, n)   F( y)   N( y)   X( y, i, n)

Proof of Theorem bnj517

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . 6 |- ( m = (/) -> ( F ` m ) = ( F ` (/) ) )
2		simpl2 1065	. . . . . . 7 |- ( ( ( N e. om /\ ph /\ ps ) /\ m e. N ) -> ph )
3		bnj517.1	. . . . . . 7 |- ( ph <-> ( F ` (/) ) = pred ( X , A , R ) )
4	2, 3	sylib 208	. . . . . 6 |- ( ( ( N e. om /\ ph /\ ps ) /\ m e. N ) -> ( F ` (/) ) = pred ( X , A , R ) )
5	1, 4	sylan9eqr 2678	. . . . 5 |- ( ( ( ( N e. om /\ ph /\ ps ) /\ m e. N ) /\ m = (/) ) -> ( F ` m ) = pred ( X , A , R ) )
6		bnj213 30952	. . . . 5 |- pred ( X , A , R ) C_ A
7	5, 6	syl6eqss 3655	. . . 4 |- ( ( ( ( N e. om /\ ph /\ ps ) /\ m e. N ) /\ m = (/) ) -> ( F ` m ) C_ A )
8		bnj517.2	. . . . . . 7 |- ( ps <-> A. i e. om ( suc i e. N -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) )
9		r19.29r 3073	. . . . . . . . . 10 |- ( ( E. i e. om m = suc i /\ A. i e. om ( suc i e. N -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) ) -> E. i e. om ( m = suc i /\ ( suc i e. N -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) ) )
10		eleq1 2689	. . . . . . . . . . . . . 14 |- ( m = suc i -> ( m e. N <-> suc i e. N ) )
11	10	biimpd 219	. . . . . . . . . . . . 13 |- ( m = suc i -> ( m e. N -> suc i e. N ) )
12		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( m = suc i -> ( F ` m ) = ( F ` suc i ) )
13	12	eqeq1d 2624	. . . . . . . . . . . . . 14 |- ( m = suc i -> ( ( F ` m ) = U_ y e. ( F ` i ) pred ( y , A , R ) <-> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) )
14		bnj213 30952	. . . . . . . . . . . . . . . . 17 |- pred ( y , A , R ) C_ A
15	14	rgenw 2924	. . . . . . . . . . . . . . . 16 |- A. y e. ( F ` i ) pred ( y , A , R ) C_ A
16		iunss 4561	. . . . . . . . . . . . . . . 16 |- ( U_ y e. ( F ` i ) pred ( y , A , R ) C_ A <-> A. y e. ( F ` i ) pred ( y , A , R ) C_ A )
17	15, 16	mpbir 221	. . . . . . . . . . . . . . 15 |- U_ y e. ( F ` i ) pred ( y , A , R ) C_ A
18		sseq1 3626	. . . . . . . . . . . . . . 15 |- ( ( F ` m ) = U_ y e. ( F ` i ) pred ( y , A , R ) -> ( ( F ` m ) C_ A <-> U_ y e. ( F ` i ) pred ( y , A , R ) C_ A ) )
19	17, 18	mpbiri 248	. . . . . . . . . . . . . 14 |- ( ( F ` m ) = U_ y e. ( F ` i ) pred ( y , A , R ) -> ( F ` m ) C_ A )
20	13, 19	syl6bir 244	. . . . . . . . . . . . 13 |- ( m = suc i -> ( ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) -> ( F ` m ) C_ A ) )
21	11, 20	imim12d 81	. . . . . . . . . . . 12 |- ( m = suc i -> ( ( suc i e. N -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) -> ( m e. N -> ( F ` m ) C_ A ) ) )
22	21	imp 445	. . . . . . . . . . 11 |- ( ( m = suc i /\ ( suc i e. N -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) ) -> ( m e. N -> ( F ` m ) C_ A ) )
23	22	rexlimivw 3029	. . . . . . . . . 10 |- ( E. i e. om ( m = suc i /\ ( suc i e. N -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) ) -> ( m e. N -> ( F ` m ) C_ A ) )
24	9, 23	syl 17	. . . . . . . . 9 |- ( ( E. i e. om m = suc i /\ A. i e. om ( suc i e. N -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) ) -> ( m e. N -> ( F ` m ) C_ A ) )
25	24	ex 450	. . . . . . . 8 |- ( E. i e. om m = suc i -> ( A. i e. om ( suc i e. N -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) -> ( m e. N -> ( F ` m ) C_ A ) ) )
26	25	com3l 89	. . . . . . 7 |- ( A. i e. om ( suc i e. N -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) -> ( m e. N -> ( E. i e. om m = suc i -> ( F ` m ) C_ A ) ) )
27	8, 26	sylbi 207	. . . . . 6 |- ( ps -> ( m e. N -> ( E. i e. om m = suc i -> ( F ` m ) C_ A ) ) )
28	27	3ad2ant3 1084	. . . . 5 |- ( ( N e. om /\ ph /\ ps ) -> ( m e. N -> ( E. i e. om m = suc i -> ( F ` m ) C_ A ) ) )
29	28	imp31 448	. . . 4 |- ( ( ( ( N e. om /\ ph /\ ps ) /\ m e. N ) /\ E. i e. om m = suc i ) -> ( F ` m ) C_ A )
30		simpr 477	. . . . . 6 |- ( ( ( N e. om /\ ph /\ ps ) /\ m e. N ) -> m e. N )
31		simpl1 1064	. . . . . 6 |- ( ( ( N e. om /\ ph /\ ps ) /\ m e. N ) -> N e. om )
32		elnn 7075	. . . . . 6 |- ( ( m e. N /\ N e. om ) -> m e. om )
33	30, 31, 32	syl2anc 693	. . . . 5 |- ( ( ( N e. om /\ ph /\ ps ) /\ m e. N ) -> m e. om )
34		nn0suc 7090	. . . . 5 |- ( m e. om -> ( m = (/) \/ E. i e. om m = suc i ) )
35	33, 34	syl 17	. . . 4 |- ( ( ( N e. om /\ ph /\ ps ) /\ m e. N ) -> ( m = (/) \/ E. i e. om m = suc i ) )
36	7, 29, 35	mpjaodan 827	. . 3 |- ( ( ( N e. om /\ ph /\ ps ) /\ m e. N ) -> ( F ` m ) C_ A )
37	36	ralrimiva 2966	. 2 |- ( ( N e. om /\ ph /\ ps ) -> A. m e. N ( F ` m ) C_ A )
38		fveq2 6191	. . . 4 |- ( m = n -> ( F ` m ) = ( F ` n ) )
39	38	sseq1d 3632	. . 3 |- ( m = n -> ( ( F ` m ) C_ A <-> ( F ` n ) C_ A ) )
40	39	cbvralv 3171	. 2 |- ( A. m e. N ( F ` m ) C_ A <-> A. n e. N ( F ` n ) C_ A )
41	37, 40	sylib 208	1 |- ( ( N e. om /\ ph /\ ps ) -> A. n e. N ( F ` n ) C_ A )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   (/)c0 3915   U_ciun 4520   suc csuc 5725   ` cfv 5888   omcom 7065   predc-bnj14 30754

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-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-rab 2921  df-v 3202  df-sbc 3436  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-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fv 5896  df-om 7066  df-bnj14 30755

This theorem is referenced by:  bnj518  30956