Theorem bnj1118 31052

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.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1118.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj1118.3	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj1118.5	|- ( ta <-> ( B e. _V /\ TrFo ( B , A , R ) /\ pred ( X , A , R ) C_ B ) )
bnj1118.7	|- D = ( om \ { (/) } )
bnj1118.18	|- ( si <-> ( ( j e. n /\ j _E i ) -> et' ) )
bnj1118.19	|- ( ph0 <-> ( i e. n /\ si /\ f e. K /\ i e. dom f ) )
bnj1118.26	|- ( et' <-> ( ( f e. K /\ j e. dom f ) -> ( f ` j ) C_ B ) )

Assertion

Ref	Expression
bnj1118	|- E. j ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( f ` i ) C_ B )

Distinct variable groups:   A, i, j, y   y, B   D, j   R, i, j, y   f, i, j, y   i, n, j

Allowed substitution hints:   ph( y, f, i, j, n)   ps( y, f, i, j, n)   ch( y, f, i, j, n)   th( y, f, i, j, n)   ta( y, f, i, j, n)   si( y, f, i, j, n)   A( f, n)   B( f, i, j, n)   D( y, f, i, n)   R( f, n)   K( y, f, i, j, n)   X( y, f, i, j, n)   et'( y, f, i, j, n)   ph0( y, f, i, j, n)

Proof of Theorem bnj1118

Step	Hyp	Ref	Expression
1		bnj1118.3	. . . 4 |- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
2		bnj1118.7	. . . 4 |- D = ( om \ { (/) } )
3		bnj1118.18	. . . 4 |- ( si <-> ( ( j e. n /\ j _E i ) -> et' ) )
4		bnj1118.19	. . . 4 |- ( ph0 <-> ( i e. n /\ si /\ f e. K /\ i e. dom f ) )
5		bnj1118.26	. . . 4 |- ( et' <-> ( ( f e. K /\ j e. dom f ) -> ( f ` j ) C_ B ) )
6	1, 2, 3, 4, 5	bnj1110 31050	. . 3 |- E. j ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( j e. n /\ i = suc j /\ ( f ` j ) C_ B ) )
7		ancl 569	. . 3 |- ( ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( j e. n /\ i = suc j /\ ( f ` j ) C_ B ) ) -> ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) /\ ( j e. n /\ i = suc j /\ ( f ` j ) C_ B ) ) ) )
8	6, 7	bnj101 30789	. 2 |- E. j ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) /\ ( j e. n /\ i = suc j /\ ( f ` j ) C_ B ) ) )
9		simpr2 1068	. . . 4 |- ( ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) /\ ( j e. n /\ i = suc j /\ ( f ` j ) C_ B ) ) -> i = suc j )
10	1	bnj1254 30880	. . . . . . 7 |- ( ch -> ps )
11	10	3ad2ant3 1084	. . . . . 6 |- ( ( th /\ ta /\ ch ) -> ps )
12	11	ad2antrl 764	. . . . 5 |- ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ps )
13	12	adantr 481	. . . 4 |- ( ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) /\ ( j e. n /\ i = suc j /\ ( f ` j ) C_ B ) ) -> ps )
14	1	bnj1232 30874	. . . . . . . . 9 |- ( ch -> n e. D )
15	14	3ad2ant3 1084	. . . . . . . 8 |- ( ( th /\ ta /\ ch ) -> n e. D )
16	15	ad2antrl 764	. . . . . . 7 |- ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> n e. D )
17	16	adantr 481	. . . . . 6 |- ( ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) /\ ( j e. n /\ i = suc j /\ ( f ` j ) C_ B ) ) -> n e. D )
18		simpr1 1067	. . . . . 6 |- ( ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) /\ ( j e. n /\ i = suc j /\ ( f ` j ) C_ B ) ) -> j e. n )
19	2	bnj923 30838	. . . . . . . 8 |- ( n e. D -> n e. om )
20	19	anim1i 592	. . . . . . 7 |- ( ( n e. D /\ j e. n ) -> ( n e. om /\ j e. n ) )
21	20	ancomd 467	. . . . . 6 |- ( ( n e. D /\ j e. n ) -> ( j e. n /\ n e. om ) )
22	17, 18, 21	syl2anc 693	. . . . 5 |- ( ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) /\ ( j e. n /\ i = suc j /\ ( f ` j ) C_ B ) ) -> ( j e. n /\ n e. om ) )
23		elnn 7075	. . . . 5 |- ( ( j e. n /\ n e. om ) -> j e. om )
24	22, 23	syl 17	. . . 4 |- ( ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) /\ ( j e. n /\ i = suc j /\ ( f ` j ) C_ B ) ) -> j e. om )
25	4	bnj1232 30874	. . . . . 6 |- ( ph0 -> i e. n )
26	25	adantl 482	. . . . 5 |- ( ( ( th /\ ta /\ ch ) /\ ph0 ) -> i e. n )
27	26	ad2antlr 763	. . . 4 |- ( ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) /\ ( j e. n /\ i = suc j /\ ( f ` j ) C_ B ) ) -> i e. n )
28	9, 13, 24, 27	bnj951 30846	. . 3 |- ( ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) /\ ( j e. n /\ i = suc j /\ ( f ` j ) C_ B ) ) -> ( i = suc j /\ ps /\ j e. om /\ i e. n ) )
29		bnj1118.5	. . . . . . 7 |- ( ta <-> ( B e. _V /\ TrFo ( B , A , R ) /\ pred ( X , A , R ) C_ B ) )
30	29	simp2bi 1077	. . . . . 6 |- ( ta -> TrFo ( B , A , R ) )
31	30	3ad2ant2 1083	. . . . 5 |- ( ( th /\ ta /\ ch ) -> TrFo ( B , A , R ) )
32	31	ad2antrl 764	. . . 4 |- ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> TrFo ( B , A , R ) )
33		simp3 1063	. . . 4 |- ( ( j e. n /\ i = suc j /\ ( f ` j ) C_ B ) -> ( f ` j ) C_ B )
34	32, 33	anim12i 590	. . 3 |- ( ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) /\ ( j e. n /\ i = suc j /\ ( f ` j ) C_ B ) ) -> ( TrFo ( B , A , R ) /\ ( f ` j ) C_ B ) )
35		bnj256 30772	. . . . 5 |- ( ( i = suc j /\ ps /\ j e. om /\ i e. n ) <-> ( ( i = suc j /\ ps ) /\ ( j e. om /\ i e. n ) ) )
36		bnj1118.2	. . . . . . . . . 10 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
37	36	bnj1112 31051	. . . . . . . . 9 |- ( ps <-> A. j ( ( j e. om /\ suc j e. n ) -> ( f ` suc j ) = U_ y e. ( f ` j ) pred ( y , A , R ) ) )
38	37	biimpi 206	. . . . . . . 8 |- ( ps -> A. j ( ( j e. om /\ suc j e. n ) -> ( f ` suc j ) = U_ y e. ( f ` j ) pred ( y , A , R ) ) )
39	38	19.21bi 2059	. . . . . . 7 |- ( ps -> ( ( j e. om /\ suc j e. n ) -> ( f ` suc j ) = U_ y e. ( f ` j ) pred ( y , A , R ) ) )
40		eleq1 2689	. . . . . . . . 9 |- ( i = suc j -> ( i e. n <-> suc j e. n ) )
41	40	anbi2d 740	. . . . . . . 8 |- ( i = suc j -> ( ( j e. om /\ i e. n ) <-> ( j e. om /\ suc j e. n ) ) )
42		fveq2 6191	. . . . . . . . 9 |- ( i = suc j -> ( f ` i ) = ( f ` suc j ) )
43	42	eqeq1d 2624	. . . . . . . 8 |- ( i = suc j -> ( ( f ` i ) = U_ y e. ( f ` j ) pred ( y , A , R ) <-> ( f ` suc j ) = U_ y e. ( f ` j ) pred ( y , A , R ) ) )
44	41, 43	imbi12d 334	. . . . . . 7 |- ( i = suc j -> ( ( ( j e. om /\ i e. n ) -> ( f ` i ) = U_ y e. ( f ` j ) pred ( y , A , R ) ) <-> ( ( j e. om /\ suc j e. n ) -> ( f ` suc j ) = U_ y e. ( f ` j ) pred ( y , A , R ) ) ) )
45	39, 44	syl5ibr 236	. . . . . 6 |- ( i = suc j -> ( ps -> ( ( j e. om /\ i e. n ) -> ( f ` i ) = U_ y e. ( f ` j ) pred ( y , A , R ) ) ) )
46	45	imp31 448	. . . . 5 |- ( ( ( i = suc j /\ ps ) /\ ( j e. om /\ i e. n ) ) -> ( f ` i ) = U_ y e. ( f ` j ) pred ( y , A , R ) )
47	35, 46	sylbi 207	. . . 4 |- ( ( i = suc j /\ ps /\ j e. om /\ i e. n ) -> ( f ` i ) = U_ y e. ( f ` j ) pred ( y , A , R ) )
48		df-bnj19 30763	. . . . . . 7 |- ( TrFo ( B , A , R ) <-> A. y e. B pred ( y , A , R ) C_ B )
49		ssralv 3666	. . . . . . 7 |- ( ( f ` j ) C_ B -> ( A. y e. B pred ( y , A , R ) C_ B -> A. y e. ( f ` j ) pred ( y , A , R ) C_ B ) )
50	48, 49	syl5bi 232	. . . . . 6 |- ( ( f ` j ) C_ B -> ( TrFo ( B , A , R ) -> A. y e. ( f ` j ) pred ( y , A , R ) C_ B ) )
51	50	impcom 446	. . . . 5 |- ( ( TrFo ( B , A , R ) /\ ( f ` j ) C_ B ) -> A. y e. ( f ` j ) pred ( y , A , R ) C_ B )
52		iunss 4561	. . . . 5 |- ( U_ y e. ( f ` j ) pred ( y , A , R ) C_ B <-> A. y e. ( f ` j ) pred ( y , A , R ) C_ B )
53	51, 52	sylibr 224	. . . 4 |- ( ( TrFo ( B , A , R ) /\ ( f ` j ) C_ B ) -> U_ y e. ( f ` j ) pred ( y , A , R ) C_ B )
54		sseq1 3626	. . . . 5 |- ( ( f ` i ) = U_ y e. ( f ` j ) pred ( y , A , R ) -> ( ( f ` i ) C_ B <-> U_ y e. ( f ` j ) pred ( y , A , R ) C_ B ) )
55	54	biimpar 502	. . . 4 |- ( ( ( f ` i ) = U_ y e. ( f ` j ) pred ( y , A , R ) /\ U_ y e. ( f ` j ) pred ( y , A , R ) C_ B ) -> ( f ` i ) C_ B )
56	47, 53, 55	syl2an 494	. . 3 |- ( ( ( i = suc j /\ ps /\ j e. om /\ i e. n ) /\ ( TrFo ( B , A , R ) /\ ( f ` j ) C_ B ) ) -> ( f ` i ) C_ B )
57	28, 34, 56	syl2anc 693	. 2 |- ( ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) /\ ( j e. n /\ i = suc j /\ ( f ` j ) C_ B ) ) -> ( f ` i ) C_ B )
58	8, 57	bnj1023 30851	1 |- E. j ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( f ` i ) C_ B )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   \ cdif 3571   C_ wss 3574   (/)c0 3915   {csn 4177   U_ciun 4520   class class class wbr 4653   _E cep 5028   dom cdm 5114   suc csuc 5725   Fn wfn 5883   ` cfv 5888   omcom 7065   /\ w-bnj17 30752   predc-bnj14 30754   TrFow-bnj19 30762

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-fn 5891  df-fv 5896  df-om 7066  df-bnj17 30753  df-bnj19 30763

This theorem is referenced by:  bnj1030  31055