Theorem bnj1110 31050

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
bnj1110.3	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj1110.7	|- D = ( om \ { (/) } )
bnj1110.18	|- ( si <-> ( ( j e. n /\ j _E i ) -> et' ) )
bnj1110.19	|- ( ph0 <-> ( i e. n /\ si /\ f e. K /\ i e. dom f ) )
bnj1110.26	|- ( et' <-> ( ( f e. K /\ j e. dom f ) -> ( f ` j ) C_ B ) )

Assertion

Ref	Expression
bnj1110	|- E. j ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( j e. n /\ i = suc j /\ ( f ` j ) C_ B ) )

Distinct variable groups:   D, j   i, j   j, n

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

Proof of Theorem bnj1110

Step	Hyp	Ref	Expression
1		bnj1110.7	. . . . . . . . 9 |- D = ( om \ { (/) } )
2	1	bnj1098 30854	. . . . . . . 8 |- E. j ( ( i =/= (/) /\ i e. n /\ n e. D ) -> ( j e. n /\ i = suc j ) )
3		bnj219 30801	. . . . . . . . . . 11 |- ( i = suc j -> j _E i )
4	3	adantl 482	. . . . . . . . . 10 |- ( ( j e. n /\ i = suc j ) -> j _E i )
5	4	ancli 574	. . . . . . . . 9 |- ( ( j e. n /\ i = suc j ) -> ( ( j e. n /\ i = suc j ) /\ j _E i ) )
6		df-3an 1039	. . . . . . . . 9 |- ( ( j e. n /\ i = suc j /\ j _E i ) <-> ( ( j e. n /\ i = suc j ) /\ j _E i ) )
7	5, 6	sylibr 224	. . . . . . . 8 |- ( ( j e. n /\ i = suc j ) -> ( j e. n /\ i = suc j /\ j _E i ) )
8	2, 7	bnj1023 30851	. . . . . . 7 |- E. j ( ( i =/= (/) /\ i e. n /\ n e. D ) -> ( j e. n /\ i = suc j /\ j _E i ) )
9		bnj1110.3	. . . . . . . . . . . 12 |- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
10	9	bnj1232 30874	. . . . . . . . . . 11 |- ( ch -> n e. D )
11	10	3ad2ant3 1084	. . . . . . . . . 10 |- ( ( th /\ ta /\ ch ) -> n e. D )
12		bnj1110.19	. . . . . . . . . . 11 |- ( ph0 <-> ( i e. n /\ si /\ f e. K /\ i e. dom f ) )
13	12	bnj1232 30874	. . . . . . . . . 10 |- ( ph0 -> i e. n )
14	11, 13	anim12ci 591	. . . . . . . . 9 |- ( ( ( th /\ ta /\ ch ) /\ ph0 ) -> ( i e. n /\ n e. D ) )
15	14	anim2i 593	. . . . . . . 8 |- ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( i =/= (/) /\ ( i e. n /\ n e. D ) ) )
16		3anass 1042	. . . . . . . 8 |- ( ( i =/= (/) /\ i e. n /\ n e. D ) <-> ( i =/= (/) /\ ( i e. n /\ n e. D ) ) )
17	15, 16	sylibr 224	. . . . . . 7 |- ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( i =/= (/) /\ i e. n /\ n e. D ) )
18	8, 17	bnj1101 30855	. . . . . 6 |- E. j ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( j e. n /\ i = suc j /\ j _E i ) )
19		3simpb 1059	. . . . . . . . 9 |- ( ( j e. n /\ i = suc j /\ j _E i ) -> ( j e. n /\ j _E i ) )
20	12	bnj1235 30875	. . . . . . . . . . 11 |- ( ph0 -> si )
21	20	ad2antll 765	. . . . . . . . . 10 |- ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> si )
22		bnj1110.18	. . . . . . . . . 10 |- ( si <-> ( ( j e. n /\ j _E i ) -> et' ) )
23	21, 22	sylib 208	. . . . . . . . 9 |- ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( ( j e. n /\ j _E i ) -> et' ) )
24	19, 23	syl5 34	. . . . . . . 8 |- ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( ( j e. n /\ i = suc j /\ j _E i ) -> et' ) )
25	24	a2i 14	. . . . . . 7 |- ( ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( j e. n /\ i = suc j /\ j _E i ) ) -> ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> et' ) )
26		pm3.43 906	. . . . . . 7 |- ( ( ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( j e. n /\ i = suc j /\ j _E i ) ) /\ ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> et' ) ) -> ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( ( j e. n /\ i = suc j /\ j _E i ) /\ et' ) ) )
27	25, 26	mpdan 702	. . . . . 6 |- ( ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( j e. n /\ i = suc j /\ j _E i ) ) -> ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( ( j e. n /\ i = suc j /\ j _E i ) /\ et' ) ) )
28	18, 27	bnj101 30789	. . . . 5 |- E. j ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( ( j e. n /\ i = suc j /\ j _E i ) /\ et' ) )
29	12	bnj1247 30879	. . . . . . 7 |- ( ph0 -> f e. K )
30	29	ad2antll 765	. . . . . 6 |- ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> f e. K )
31		pm3.43i 472	. . . . . 6 |- ( ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> f e. K ) -> ( ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( ( j e. n /\ i = suc j /\ j _E i ) /\ et' ) ) -> ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( f e. K /\ ( ( j e. n /\ i = suc j /\ j _E i ) /\ et' ) ) ) ) )
32	30, 31	ax-mp 5	. . . . 5 |- ( ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( ( j e. n /\ i = suc j /\ j _E i ) /\ et' ) ) -> ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( f e. K /\ ( ( j e. n /\ i = suc j /\ j _E i ) /\ et' ) ) ) )
33	28, 32	bnj101 30789	. . . 4 |- E. j ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( f e. K /\ ( ( j e. n /\ i = suc j /\ j _E i ) /\ et' ) ) )
34		fndm 5990	. . . . . . . . 9 |- ( f Fn n -> dom f = n )
35	9, 34	bnj770 30833	. . . . . . . 8 |- ( ch -> dom f = n )
36	35	3ad2ant3 1084	. . . . . . 7 |- ( ( th /\ ta /\ ch ) -> dom f = n )
37	36	ad2antrl 764	. . . . . 6 |- ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> dom f = n )
38	37	eleq2d 2687	. . . . 5 |- ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( j e. dom f <-> j e. n ) )
39		pm3.43i 472	. . . . 5 |- ( ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( j e. dom f <-> j e. n ) ) -> ( ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( f e. K /\ ( ( j e. n /\ i = suc j /\ j _E i ) /\ et' ) ) ) -> ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( ( j e. dom f <-> j e. n ) /\ ( f e. K /\ ( ( j e. n /\ i = suc j /\ j _E i ) /\ et' ) ) ) ) ) )
40	38, 39	ax-mp 5	. . . 4 |- ( ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( f e. K /\ ( ( j e. n /\ i = suc j /\ j _E i ) /\ et' ) ) ) -> ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( ( j e. dom f <-> j e. n ) /\ ( f e. K /\ ( ( j e. n /\ i = suc j /\ j _E i ) /\ et' ) ) ) ) )
41	33, 40	bnj101 30789	. . 3 |- E. j ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( ( j e. dom f <-> j e. n ) /\ ( f e. K /\ ( ( j e. n /\ i = suc j /\ j _E i ) /\ et' ) ) ) )
42		bnj268 30775	. . . . . 6 |- ( ( ( j e. dom f <-> j e. n ) /\ f e. K /\ ( j e. n /\ i = suc j /\ j _E i ) /\ et' ) <-> ( ( j e. dom f <-> j e. n ) /\ ( j e. n /\ i = suc j /\ j _E i ) /\ f e. K /\ et' ) )
43		bnj251 30768	. . . . . 6 |- ( ( ( j e. dom f <-> j e. n ) /\ f e. K /\ ( j e. n /\ i = suc j /\ j _E i ) /\ et' ) <-> ( ( j e. dom f <-> j e. n ) /\ ( f e. K /\ ( ( j e. n /\ i = suc j /\ j _E i ) /\ et' ) ) ) )
44	42, 43	bitr3i 266	. . . . 5 |- ( ( ( j e. dom f <-> j e. n ) /\ ( j e. n /\ i = suc j /\ j _E i ) /\ f e. K /\ et' ) <-> ( ( j e. dom f <-> j e. n ) /\ ( f e. K /\ ( ( j e. n /\ i = suc j /\ j _E i ) /\ et' ) ) ) )
45	44	imbi2i 326	. . . 4 |- ( ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( ( j e. dom f <-> j e. n ) /\ ( j e. n /\ i = suc j /\ j _E i ) /\ f e. K /\ et' ) ) <-> ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( ( j e. dom f <-> j e. n ) /\ ( f e. K /\ ( ( j e. n /\ i = suc j /\ j _E i ) /\ et' ) ) ) ) )
46	45	exbii 1774	. . 3 |- ( E. j ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( ( j e. dom f <-> j e. n ) /\ ( j e. n /\ i = suc j /\ j _E i ) /\ f e. K /\ et' ) ) <-> E. j ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( ( j e. dom f <-> j e. n ) /\ ( f e. K /\ ( ( j e. n /\ i = suc j /\ j _E i ) /\ et' ) ) ) ) )
47	41, 46	mpbir 221	. 2 |- E. j ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( ( j e. dom f <-> j e. n ) /\ ( j e. n /\ i = suc j /\ j _E i ) /\ f e. K /\ et' ) )
48		simp1 1061	. . . 4 |- ( ( j e. n /\ i = suc j /\ j _E i ) -> j e. n )
49	48	bnj706 30824	. . 3 |- ( ( ( j e. dom f <-> j e. n ) /\ ( j e. n /\ i = suc j /\ j _E i ) /\ f e. K /\ et' ) -> j e. n )
50		simp2 1062	. . . 4 |- ( ( j e. n /\ i = suc j /\ j _E i ) -> i = suc j )
51	50	bnj706 30824	. . 3 |- ( ( ( j e. dom f <-> j e. n ) /\ ( j e. n /\ i = suc j /\ j _E i ) /\ f e. K /\ et' ) -> i = suc j )
52		bnj258 30774	. . . . 5 |- ( ( ( j e. dom f <-> j e. n ) /\ ( j e. n /\ i = suc j /\ j _E i ) /\ f e. K /\ et' ) <-> ( ( ( j e. dom f <-> j e. n ) /\ ( j e. n /\ i = suc j /\ j _E i ) /\ et' ) /\ f e. K ) )
53	52	simprbi 480	. . . 4 |- ( ( ( j e. dom f <-> j e. n ) /\ ( j e. n /\ i = suc j /\ j _E i ) /\ f e. K /\ et' ) -> f e. K )
54		bnj642 30818	. . . . 5 |- ( ( ( j e. dom f <-> j e. n ) /\ ( j e. n /\ i = suc j /\ j _E i ) /\ f e. K /\ et' ) -> ( j e. dom f <-> j e. n ) )
55	49, 54	mpbird 247	. . . 4 |- ( ( ( j e. dom f <-> j e. n ) /\ ( j e. n /\ i = suc j /\ j _E i ) /\ f e. K /\ et' ) -> j e. dom f )
56		bnj645 30820	. . . . 5 |- ( ( ( j e. dom f <-> j e. n ) /\ ( j e. n /\ i = suc j /\ j _E i ) /\ f e. K /\ et' ) -> et' )
57		bnj1110.26	. . . . 5 |- ( et' <-> ( ( f e. K /\ j e. dom f ) -> ( f ` j ) C_ B ) )
58	56, 57	sylib 208	. . . 4 |- ( ( ( j e. dom f <-> j e. n ) /\ ( j e. n /\ i = suc j /\ j _E i ) /\ f e. K /\ et' ) -> ( ( f e. K /\ j e. dom f ) -> ( f ` j ) C_ B ) )
59	53, 55, 58	mp2and 715	. . 3 |- ( ( ( j e. dom f <-> j e. n ) /\ ( j e. n /\ i = suc j /\ j _E i ) /\ f e. K /\ et' ) -> ( f ` j ) C_ B )
60	49, 51, 59	3jca 1242	. 2 |- ( ( ( j e. dom f <-> j e. n ) /\ ( j e. n /\ i = suc j /\ j _E i ) /\ f e. K /\ et' ) -> ( j e. n /\ i = suc j /\ ( f ` j ) C_ B ) )
61	47, 60	bnj1023 30851	1 |- E. j ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( j e. n /\ i = suc j /\ ( f ` j ) C_ B ) )

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

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-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-fn 5891  df-om 7066  df-bnj17 30753

This theorem is referenced by:  bnj1118  31052