Theorem bnj594 30982

Description: Technical lemma for bnj852 30991. 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
bnj594.1	|- ( ph <-> ( f ` (/) ) = pred ( x , A , R ) )
bnj594.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj594.3	|- ( ch <-> ( f Fn n /\ ph /\ ps ) )
bnj594.7	|- D = ( om \ { (/) } )
bnj594.9	|- ( ph' <-> ( g ` (/) ) = pred ( x , A , R ) )
bnj594.10	|- ( ps' <-> A. i e. om ( suc i e. n -> ( g ` suc i ) = U_ y e. ( g ` i ) pred ( y , A , R ) ) )
bnj594.11	|- ( ch' <-> ( g Fn n /\ ph' /\ ps' ) )
bnj594.15	|- ( th <-> ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) )
bnj594.16	|- ( [. k / j ]. th <-> ( ( n e. D /\ ch /\ ch' ) -> ( f ` k ) = ( g ` k ) ) )
bnj594.17	|- ( ta <-> A. k e. n ( k _E j -> [. k / j ]. th ) )

Assertion

Ref	Expression
bnj594	|- ( ( j e. n /\ ta ) -> th )

Distinct variable groups:   A, i, k   D, k   R, i, k   ch, k   k, ch'   f, i, k, y   g, i, k, y   i, n, k   j, k

Allowed substitution hints:   ph( x, y, f, g, i, j, k, n)   ps( x, y, f, g, i, j, k, n)   ch( x, y, f, g, i, j, n)   th( x, y, f, g, i, j, k, n)   ta( x, y, f, g, i, j, k, n)   A( x, y, f, g, j, n)   D( x, y, f, g, i, j, n)   R( x, y, f, g, j, n)   ph'( x, y, f, g, i, j, k, n)   ps'( x, y, f, g, i, j, k, n)   ch'( x, y, f, g, i, j, n)

Proof of Theorem bnj594

Step	Hyp	Ref	Expression
1		bnj594.3	. . . . . . . . 9 |- ( ch <-> ( f Fn n /\ ph /\ ps ) )
2	1	simp2bi 1077	. . . . . . . 8 |- ( ch -> ph )
3		bnj594.1	. . . . . . . 8 |- ( ph <-> ( f ` (/) ) = pred ( x , A , R ) )
4	2, 3	sylib 208	. . . . . . 7 |- ( ch -> ( f ` (/) ) = pred ( x , A , R ) )
5		bnj594.11	. . . . . . . . 9 |- ( ch' <-> ( g Fn n /\ ph' /\ ps' ) )
6	5	simp2bi 1077	. . . . . . . 8 |- ( ch' -> ph' )
7		bnj594.9	. . . . . . . 8 |- ( ph' <-> ( g ` (/) ) = pred ( x , A , R ) )
8	6, 7	sylib 208	. . . . . . 7 |- ( ch' -> ( g ` (/) ) = pred ( x , A , R ) )
9		eqtr3 2643	. . . . . . 7 |- ( ( ( f ` (/) ) = pred ( x , A , R ) /\ ( g ` (/) ) = pred ( x , A , R ) ) -> ( f ` (/) ) = ( g ` (/) ) )
10	4, 8, 9	syl2an 494	. . . . . 6 |- ( ( ch /\ ch' ) -> ( f ` (/) ) = ( g ` (/) ) )
11	10	3adant1 1079	. . . . 5 |- ( ( n e. D /\ ch /\ ch' ) -> ( f ` (/) ) = ( g ` (/) ) )
12		fveq2 6191	. . . . . 6 |- ( j = (/) -> ( f ` j ) = ( f ` (/) ) )
13		fveq2 6191	. . . . . 6 |- ( j = (/) -> ( g ` j ) = ( g ` (/) ) )
14	12, 13	eqeq12d 2637	. . . . 5 |- ( j = (/) -> ( ( f ` j ) = ( g ` j ) <-> ( f ` (/) ) = ( g ` (/) ) ) )
15	11, 14	syl5ibr 236	. . . 4 |- ( j = (/) -> ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) )
16		bnj594.15	. . . 4 |- ( th <-> ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) )
17	15, 16	sylibr 224	. . 3 |- ( j = (/) -> th )
18	17	a1d 25	. 2 |- ( j = (/) -> ( ( j e. n /\ ta ) -> th ) )
19		bnj253 30770	. . . . . 6 |- ( ( n e. D /\ n e. D /\ ch /\ ch' ) <-> ( ( n e. D /\ n e. D ) /\ ch /\ ch' ) )
20		bnj252 30769	. . . . . 6 |- ( ( n e. D /\ n e. D /\ ch /\ ch' ) <-> ( n e. D /\ ( n e. D /\ ch /\ ch' ) ) )
21		anidm 676	. . . . . . 7 |- ( ( n e. D /\ n e. D ) <-> n e. D )
22	21	3anbi1i 1253	. . . . . 6 |- ( ( ( n e. D /\ n e. D ) /\ ch /\ ch' ) <-> ( n e. D /\ ch /\ ch' ) )
23	19, 20, 22	3bitr3i 290	. . . . 5 |- ( ( n e. D /\ ( n e. D /\ ch /\ ch' ) ) <-> ( n e. D /\ ch /\ ch' ) )
24		df-bnj17 30753	. . . . . . . . . 10 |- ( ( j =/= (/) /\ j e. n /\ n e. D /\ ta ) <-> ( ( j =/= (/) /\ j e. n /\ n e. D ) /\ ta ) )
25		bnj594.17	. . . . . . . . . . . 12 |- ( ta <-> A. k e. n ( k _E j -> [. k / j ]. th ) )
26	25	bnj1095 30852	. . . . . . . . . . 11 |- ( ta -> A. k ta )
27	26	bnj1352 30898	. . . . . . . . . 10 |- ( ( ( j =/= (/) /\ j e. n /\ n e. D ) /\ ta ) -> A. k ( ( j =/= (/) /\ j e. n /\ n e. D ) /\ ta ) )
28	24, 27	hbxfrbi 1752	. . . . . . . . 9 |- ( ( j =/= (/) /\ j e. n /\ n e. D /\ ta ) -> A. k ( j =/= (/) /\ j e. n /\ n e. D /\ ta ) )
29		bnj170 30764	. . . . . . . . . . . 12 |- ( ( j =/= (/) /\ j e. n /\ n e. D ) <-> ( ( j e. n /\ n e. D ) /\ j =/= (/) ) )
30		bnj594.7	. . . . . . . . . . . . . . 15 |- D = ( om \ { (/) } )
31	30	bnj923 30838	. . . . . . . . . . . . . 14 |- ( n e. D -> n e. om )
32		elnn 7075	. . . . . . . . . . . . . 14 |- ( ( j e. n /\ n e. om ) -> j e. om )
33	31, 32	sylan2 491	. . . . . . . . . . . . 13 |- ( ( j e. n /\ n e. D ) -> j e. om )
34	33	anim1i 592	. . . . . . . . . . . 12 |- ( ( ( j e. n /\ n e. D ) /\ j =/= (/) ) -> ( j e. om /\ j =/= (/) ) )
35	29, 34	sylbi 207	. . . . . . . . . . 11 |- ( ( j =/= (/) /\ j e. n /\ n e. D ) -> ( j e. om /\ j =/= (/) ) )
36		nnsuc 7082	. . . . . . . . . . 11 |- ( ( j e. om /\ j =/= (/) ) -> E. k e. om j = suc k )
37		rexex 3002	. . . . . . . . . . 11 |- ( E. k e. om j = suc k -> E. k j = suc k )
38	35, 36, 37	3syl 18	. . . . . . . . . 10 |- ( ( j =/= (/) /\ j e. n /\ n e. D ) -> E. k j = suc k )
39	38	bnj721 30827	. . . . . . . . 9 |- ( ( j =/= (/) /\ j e. n /\ n e. D /\ ta ) -> E. k j = suc k )
40	28, 39	bnj596 30816	. . . . . . . 8 |- ( ( j =/= (/) /\ j e. n /\ n e. D /\ ta ) -> E. k ( ( j =/= (/) /\ j e. n /\ n e. D /\ ta ) /\ j = suc k ) )
41		bnj667 30822	. . . . . . . . . . 11 |- ( ( j =/= (/) /\ j e. n /\ n e. D /\ ta ) -> ( j e. n /\ n e. D /\ ta ) )
42	41	anim1i 592	. . . . . . . . . 10 |- ( ( ( j =/= (/) /\ j e. n /\ n e. D /\ ta ) /\ j = suc k ) -> ( ( j e. n /\ n e. D /\ ta ) /\ j = suc k ) )
43		bnj258 30774	. . . . . . . . . 10 |- ( ( j e. n /\ n e. D /\ j = suc k /\ ta ) <-> ( ( j e. n /\ n e. D /\ ta ) /\ j = suc k ) )
44	42, 43	sylibr 224	. . . . . . . . 9 |- ( ( ( j =/= (/) /\ j e. n /\ n e. D /\ ta ) /\ j = suc k ) -> ( j e. n /\ n e. D /\ j = suc k /\ ta ) )
45		df-bnj17 30753	. . . . . . . . . . . . . . 15 |- ( ( j e. n /\ n e. D /\ j = suc k /\ ta ) <-> ( ( j e. n /\ n e. D /\ j = suc k ) /\ ta ) )
46		bnj219 30801	. . . . . . . . . . . . . . . . . 18 |- ( j = suc k -> k _E j )
47	46	3ad2ant3 1084	. . . . . . . . . . . . . . . . 17 |- ( ( j e. n /\ n e. D /\ j = suc k ) -> k _E j )
48	47	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ( j e. n /\ n e. D /\ j = suc k ) /\ ta ) -> k _E j )
49		vex 3203	. . . . . . . . . . . . . . . . . . 19 |- k e. _V
50	49	bnj216 30800	. . . . . . . . . . . . . . . . . 18 |- ( j = suc k -> k e. j )
51		df-3an 1039	. . . . . . . . . . . . . . . . . . . 20 |- ( ( k e. j /\ j e. n /\ n e. D ) <-> ( ( k e. j /\ j e. n ) /\ n e. D ) )
52		3anrot 1043	. . . . . . . . . . . . . . . . . . . 20 |- ( ( k e. j /\ j e. n /\ n e. D ) <-> ( j e. n /\ n e. D /\ k e. j ) )
53		ancom 466	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( k e. j /\ j e. n ) /\ n e. D ) <-> ( n e. D /\ ( k e. j /\ j e. n ) ) )
54	51, 52, 53	3bitr3i 290	. . . . . . . . . . . . . . . . . . 19 |- ( ( j e. n /\ n e. D /\ k e. j ) <-> ( n e. D /\ ( k e. j /\ j e. n ) ) )
55		eldifi 3732	. . . . . . . . . . . . . . . . . . . . . 22 |- ( n e. ( om \ { (/) } ) -> n e. om )
56	55, 30	eleq2s 2719	. . . . . . . . . . . . . . . . . . . . 21 |- ( n e. D -> n e. om )
57		nnord 7073	. . . . . . . . . . . . . . . . . . . . 21 |- ( n e. om -> Ord n )
58		ordtr1 5767	. . . . . . . . . . . . . . . . . . . . 21 |- ( Ord n -> ( ( k e. j /\ j e. n ) -> k e. n ) )
59	56, 57, 58	3syl 18	. . . . . . . . . . . . . . . . . . . 20 |- ( n e. D -> ( ( k e. j /\ j e. n ) -> k e. n ) )
60	59	imp 445	. . . . . . . . . . . . . . . . . . 19 |- ( ( n e. D /\ ( k e. j /\ j e. n ) ) -> k e. n )
61	54, 60	sylbi 207	. . . . . . . . . . . . . . . . . 18 |- ( ( j e. n /\ n e. D /\ k e. j ) -> k e. n )
62	50, 61	syl3an3 1361	. . . . . . . . . . . . . . . . 17 |- ( ( j e. n /\ n e. D /\ j = suc k ) -> k e. n )
63		rsp 2929	. . . . . . . . . . . . . . . . . 18 |- ( A. k e. n ( k _E j -> [. k / j ]. th ) -> ( k e. n -> ( k _E j -> [. k / j ]. th ) ) )
64	25, 63	sylbi 207	. . . . . . . . . . . . . . . . 17 |- ( ta -> ( k e. n -> ( k _E j -> [. k / j ]. th ) ) )
65	62, 64	mpan9 486	. . . . . . . . . . . . . . . 16 |- ( ( ( j e. n /\ n e. D /\ j = suc k ) /\ ta ) -> ( k _E j -> [. k / j ]. th ) )
66	48, 65	mpd 15	. . . . . . . . . . . . . . 15 |- ( ( ( j e. n /\ n e. D /\ j = suc k ) /\ ta ) -> [. k / j ]. th )
67	45, 66	sylbi 207	. . . . . . . . . . . . . 14 |- ( ( j e. n /\ n e. D /\ j = suc k /\ ta ) -> [. k / j ]. th )
68	67	anim1i 592	. . . . . . . . . . . . 13 |- ( ( ( j e. n /\ n e. D /\ j = suc k /\ ta ) /\ ( n e. D /\ ch /\ ch' ) ) -> ( [. k / j ]. th /\ ( n e. D /\ ch /\ ch' ) ) )
69		bnj252 30769	. . . . . . . . . . . . 13 |- ( ( [. k / j ]. th /\ n e. D /\ ch /\ ch' ) <-> ( [. k / j ]. th /\ ( n e. D /\ ch /\ ch' ) ) )
70	68, 69	sylibr 224	. . . . . . . . . . . 12 |- ( ( ( j e. n /\ n e. D /\ j = suc k /\ ta ) /\ ( n e. D /\ ch /\ ch' ) ) -> ( [. k / j ]. th /\ n e. D /\ ch /\ ch' ) )
71		bnj446 30783	. . . . . . . . . . . . 13 |- ( ( [. k / j ]. th /\ n e. D /\ ch /\ ch' ) <-> ( ( n e. D /\ ch /\ ch' ) /\ [. k / j ]. th ) )
72		bnj594.16	. . . . . . . . . . . . . 14 |- ( [. k / j ]. th <-> ( ( n e. D /\ ch /\ ch' ) -> ( f ` k ) = ( g ` k ) ) )
73		pm3.35 611	. . . . . . . . . . . . . 14 |- ( ( ( n e. D /\ ch /\ ch' ) /\ ( ( n e. D /\ ch /\ ch' ) -> ( f ` k ) = ( g ` k ) ) ) -> ( f ` k ) = ( g ` k ) )
74	72, 73	sylan2b 492	. . . . . . . . . . . . 13 |- ( ( ( n e. D /\ ch /\ ch' ) /\ [. k / j ]. th ) -> ( f ` k ) = ( g ` k ) )
75	71, 74	sylbi 207	. . . . . . . . . . . 12 |- ( ( [. k / j ]. th /\ n e. D /\ ch /\ ch' ) -> ( f ` k ) = ( g ` k ) )
76		iuneq1 4534	. . . . . . . . . . . 12 |- ( ( f ` k ) = ( g ` k ) -> U_ y e. ( f ` k ) pred ( y , A , R ) = U_ y e. ( g ` k ) pred ( y , A , R ) )
77	70, 75, 76	3syl 18	. . . . . . . . . . 11 |- ( ( ( j e. n /\ n e. D /\ j = suc k /\ ta ) /\ ( n e. D /\ ch /\ ch' ) ) -> U_ y e. ( f ` k ) pred ( y , A , R ) = U_ y e. ( g ` k ) pred ( y , A , R ) )
78		bnj658 30821	. . . . . . . . . . . . 13 |- ( ( j e. n /\ n e. D /\ j = suc k /\ ta ) -> ( j e. n /\ n e. D /\ j = suc k ) )
79	1	simp3bi 1078	. . . . . . . . . . . . . 14 |- ( ch -> ps )
80	5	simp3bi 1078	. . . . . . . . . . . . . 14 |- ( ch' -> ps' )
81	79, 80	bnj240 30765	. . . . . . . . . . . . 13 |- ( ( n e. D /\ ch /\ ch' ) -> ( ps /\ ps' ) )
82	78, 81	anim12i 590	. . . . . . . . . . . 12 |- ( ( ( j e. n /\ n e. D /\ j = suc k /\ ta ) /\ ( n e. D /\ ch /\ ch' ) ) -> ( ( j e. n /\ n e. D /\ j = suc k ) /\ ( ps /\ ps' ) ) )
83		simpl 473	. . . . . . . . . . . . 13 |- ( ( ps /\ ps' ) -> ps )
84	83	anim2i 593	. . . . . . . . . . . 12 |- ( ( ( j e. n /\ n e. D /\ j = suc k ) /\ ( ps /\ ps' ) ) -> ( ( j e. n /\ n e. D /\ j = suc k ) /\ ps ) )
85		simp3 1063	. . . . . . . . . . . . . 14 |- ( ( j e. n /\ n e. D /\ j = suc k ) -> j = suc k )
86	85	anim1i 592	. . . . . . . . . . . . 13 |- ( ( ( j e. n /\ n e. D /\ j = suc k ) /\ ps ) -> ( j = suc k /\ ps ) )
87		simpl1 1064	. . . . . . . . . . . . . 14 |- ( ( ( j e. n /\ n e. D /\ j = suc k ) /\ ( j = suc k /\ ps ) ) -> j e. n )
88		df-3an 1039	. . . . . . . . . . . . . . . . 17 |- ( ( j e. n /\ n e. D /\ j = suc k ) <-> ( ( j e. n /\ n e. D ) /\ j = suc k ) )
89		ancom 466	. . . . . . . . . . . . . . . . 17 |- ( ( ( j e. n /\ n e. D ) /\ j = suc k ) <-> ( j = suc k /\ ( j e. n /\ n e. D ) ) )
90	88, 89	bitri 264	. . . . . . . . . . . . . . . 16 |- ( ( j e. n /\ n e. D /\ j = suc k ) <-> ( j = suc k /\ ( j e. n /\ n e. D ) ) )
91		elnn 7075	. . . . . . . . . . . . . . . . 17 |- ( ( k e. j /\ j e. om ) -> k e. om )
92	50, 33, 91	syl2an 494	. . . . . . . . . . . . . . . 16 |- ( ( j = suc k /\ ( j e. n /\ n e. D ) ) -> k e. om )
93	90, 92	sylbi 207	. . . . . . . . . . . . . . 15 |- ( ( j e. n /\ n e. D /\ j = suc k ) -> k e. om )
94		bnj594.2	. . . . . . . . . . . . . . . . 17 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
95	94	bnj589 30979	. . . . . . . . . . . . . . . 16 |- ( ps <-> A. k e. om ( suc k e. n -> ( f ` suc k ) = U_ y e. ( f ` k ) pred ( y , A , R ) ) )
96	95	bnj590 30980	. . . . . . . . . . . . . . 15 |- ( ( j = suc k /\ ps ) -> ( k e. om -> ( j e. n -> ( f ` j ) = U_ y e. ( f ` k ) pred ( y , A , R ) ) ) )
97	93, 96	mpan9 486	. . . . . . . . . . . . . 14 |- ( ( ( j e. n /\ n e. D /\ j = suc k ) /\ ( j = suc k /\ ps ) ) -> ( j e. n -> ( f ` j ) = U_ y e. ( f ` k ) pred ( y , A , R ) ) )
98	87, 97	mpd 15	. . . . . . . . . . . . 13 |- ( ( ( j e. n /\ n e. D /\ j = suc k ) /\ ( j = suc k /\ ps ) ) -> ( f ` j ) = U_ y e. ( f ` k ) pred ( y , A , R ) )
99	86, 98	syldan 487	. . . . . . . . . . . 12 |- ( ( ( j e. n /\ n e. D /\ j = suc k ) /\ ps ) -> ( f ` j ) = U_ y e. ( f ` k ) pred ( y , A , R ) )
100	82, 84, 99	3syl 18	. . . . . . . . . . 11 |- ( ( ( j e. n /\ n e. D /\ j = suc k /\ ta ) /\ ( n e. D /\ ch /\ ch' ) ) -> ( f ` j ) = U_ y e. ( f ` k ) pred ( y , A , R ) )
101		simpr 477	. . . . . . . . . . . . 13 |- ( ( ps /\ ps' ) -> ps' )
102	101	anim2i 593	. . . . . . . . . . . 12 |- ( ( ( j e. n /\ n e. D /\ j = suc k ) /\ ( ps /\ ps' ) ) -> ( ( j e. n /\ n e. D /\ j = suc k ) /\ ps' ) )
103	85	anim1i 592	. . . . . . . . . . . . 13 |- ( ( ( j e. n /\ n e. D /\ j = suc k ) /\ ps' ) -> ( j = suc k /\ ps' ) )
104		simpl1 1064	. . . . . . . . . . . . . 14 |- ( ( ( j e. n /\ n e. D /\ j = suc k ) /\ ( j = suc k /\ ps' ) ) -> j e. n )
105		bnj594.10	. . . . . . . . . . . . . . . . 17 |- ( ps' <-> A. i e. om ( suc i e. n -> ( g ` suc i ) = U_ y e. ( g ` i ) pred ( y , A , R ) ) )
106	105	bnj589 30979	. . . . . . . . . . . . . . . 16 |- ( ps' <-> A. k e. om ( suc k e. n -> ( g ` suc k ) = U_ y e. ( g ` k ) pred ( y , A , R ) ) )
107	106	bnj590 30980	. . . . . . . . . . . . . . 15 |- ( ( j = suc k /\ ps' ) -> ( k e. om -> ( j e. n -> ( g ` j ) = U_ y e. ( g ` k ) pred ( y , A , R ) ) ) )
108	93, 107	mpan9 486	. . . . . . . . . . . . . 14 |- ( ( ( j e. n /\ n e. D /\ j = suc k ) /\ ( j = suc k /\ ps' ) ) -> ( j e. n -> ( g ` j ) = U_ y e. ( g ` k ) pred ( y , A , R ) ) )
109	104, 108	mpd 15	. . . . . . . . . . . . 13 |- ( ( ( j e. n /\ n e. D /\ j = suc k ) /\ ( j = suc k /\ ps' ) ) -> ( g ` j ) = U_ y e. ( g ` k ) pred ( y , A , R ) )
110	103, 109	syldan 487	. . . . . . . . . . . 12 |- ( ( ( j e. n /\ n e. D /\ j = suc k ) /\ ps' ) -> ( g ` j ) = U_ y e. ( g ` k ) pred ( y , A , R ) )
111	82, 102, 110	3syl 18	. . . . . . . . . . 11 |- ( ( ( j e. n /\ n e. D /\ j = suc k /\ ta ) /\ ( n e. D /\ ch /\ ch' ) ) -> ( g ` j ) = U_ y e. ( g ` k ) pred ( y , A , R ) )
112	77, 100, 111	3eqtr4d 2666	. . . . . . . . . 10 |- ( ( ( j e. n /\ n e. D /\ j = suc k /\ ta ) /\ ( n e. D /\ ch /\ ch' ) ) -> ( f ` j ) = ( g ` j ) )
113	112	ex 450	. . . . . . . . 9 |- ( ( j e. n /\ n e. D /\ j = suc k /\ ta ) -> ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) )
114	44, 113	syl 17	. . . . . . . 8 |- ( ( ( j =/= (/) /\ j e. n /\ n e. D /\ ta ) /\ j = suc k ) -> ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) )
115	40, 114	bnj593 30815	. . . . . . 7 |- ( ( j =/= (/) /\ j e. n /\ n e. D /\ ta ) -> E. k ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) )
116		bnj258 30774	. . . . . . 7 |- ( ( j =/= (/) /\ j e. n /\ n e. D /\ ta ) <-> ( ( j =/= (/) /\ j e. n /\ ta ) /\ n e. D ) )
117		19.9v 1896	. . . . . . 7 |- ( E. k ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) <-> ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) )
118	115, 116, 117	3imtr3i 280	. . . . . 6 |- ( ( ( j =/= (/) /\ j e. n /\ ta ) /\ n e. D ) -> ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) )
119	118	expimpd 629	. . . . 5 |- ( ( j =/= (/) /\ j e. n /\ ta ) -> ( ( n e. D /\ ( n e. D /\ ch /\ ch' ) ) -> ( f ` j ) = ( g ` j ) ) )
120	23, 119	syl5bir 233	. . . 4 |- ( ( j =/= (/) /\ j e. n /\ ta ) -> ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) )
121	120, 16	sylibr 224	. . 3 |- ( ( j =/= (/) /\ j e. n /\ ta ) -> th )
122	121	3expib 1268	. 2 |- ( j =/= (/) -> ( ( j e. n /\ ta ) -> th ) )
123	18, 122	pm2.61ine 2877	1 |- ( ( j e. n /\ ta ) -> th )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   [.wsbc 3435   \ cdif 3571   (/)c0 3915   {csn 4177   U_ciun 4520   class class class wbr 4653   _E cep 5028   Ord word 5722   suc csuc 5725   Fn wfn 5883   ` cfv 5888   omcom 7065   /\ w-bnj17 30752   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-bnj17 30753

This theorem is referenced by:  bnj580  30983