Theorem bnj966 31014

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
bnj966.3	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj966.10	|- D = ( om \ { (/) } )
bnj966.12	|- C = U_ y e. ( f ` m ) pred ( y , A , R )
bnj966.13	|- G = ( f u. { <. n , C >. } )
bnj966.44	|- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> C e. _V )
bnj966.53	|- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> G Fn p )

Assertion

Ref	Expression
bnj966	|- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ n = suc i ) ) -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) )

Distinct variable groups:   y, f   y, i   y, m   y, n

Allowed substitution hints:   ph( y, f, i, m, n, p)   ps( y, f, i, m, n, p)   ch( y, f, i, m, n, p)   A( y, f, i, m, n, p)   C( y, f, i, m, n, p)   D( y, f, i, m, n, p)   R( y, f, i, m, n, p)   G( y, f, i, m, n, p)   X( y, f, i, m, n, p)

Proof of Theorem bnj966

Step	Hyp	Ref	Expression
1		bnj966.53	. . . . . 6 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> G Fn p )
2	1	bnj930 30840	. . . . 5 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> Fun G )
3	2	3adant3 1081	. . . 4 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ n = suc i ) ) -> Fun G )
4		opex 4932	. . . . . . 7 |- <. n , C >. e. _V
5	4	snid 4208	. . . . . 6 |- <. n , C >. e. { <. n , C >. }
6		elun2 3781	. . . . . 6 |- ( <. n , C >. e. { <. n , C >. } -> <. n , C >. e. ( f u. { <. n , C >. } ) )
7	5, 6	ax-mp 5	. . . . 5 |- <. n , C >. e. ( f u. { <. n , C >. } )
8		bnj966.13	. . . . 5 |- G = ( f u. { <. n , C >. } )
9	7, 8	eleqtrri 2700	. . . 4 |- <. n , C >. e. G
10		funopfv 6235	. . . 4 |- ( Fun G -> ( <. n , C >. e. G -> ( G ` n ) = C ) )
11	3, 9, 10	mpisyl 21	. . 3 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ n = suc i ) ) -> ( G ` n ) = C )
12		simp22 1095	. . . 4 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ n = suc i ) ) -> n = suc m )
13		simp33 1099	. . . . 5 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ n = suc i ) ) -> n = suc i )
14		bnj551 30812	. . . . 5 |- ( ( n = suc m /\ n = suc i ) -> m = i )
15	12, 13, 14	syl2anc 693	. . . 4 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ n = suc i ) ) -> m = i )
16		suceq 5790	. . . . . . . 8 |- ( m = i -> suc m = suc i )
17	16	eqeq2d 2632	. . . . . . 7 |- ( m = i -> ( n = suc m <-> n = suc i ) )
18	17	biimpac 503	. . . . . 6 |- ( ( n = suc m /\ m = i ) -> n = suc i )
19	18	fveq2d 6195	. . . . 5 |- ( ( n = suc m /\ m = i ) -> ( G ` n ) = ( G ` suc i ) )
20		bnj966.12	. . . . . . 7 |- C = U_ y e. ( f ` m ) pred ( y , A , R )
21		fveq2 6191	. . . . . . . 8 |- ( m = i -> ( f ` m ) = ( f ` i ) )
22	21	bnj1113 30856	. . . . . . 7 |- ( m = i -> U_ y e. ( f ` m ) pred ( y , A , R ) = U_ y e. ( f ` i ) pred ( y , A , R ) )
23	20, 22	syl5eq 2668	. . . . . 6 |- ( m = i -> C = U_ y e. ( f ` i ) pred ( y , A , R ) )
24	23	adantl 482	. . . . 5 |- ( ( n = suc m /\ m = i ) -> C = U_ y e. ( f ` i ) pred ( y , A , R ) )
25	19, 24	eqeq12d 2637	. . . 4 |- ( ( n = suc m /\ m = i ) -> ( ( G ` n ) = C <-> ( G ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
26	12, 15, 25	syl2anc 693	. . 3 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ n = suc i ) ) -> ( ( G ` n ) = C <-> ( G ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
27	11, 26	mpbid 222	. 2 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ n = suc i ) ) -> ( G ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) )
28		bnj966.44	. . . . . 6 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> C e. _V )
29	28	3adant3 1081	. . . . 5 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ n = suc i ) ) -> C e. _V )
30		bnj966.3	. . . . . . . 8 |- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
31	30	bnj1235 30875	. . . . . . 7 |- ( ch -> f Fn n )
32	31	3ad2ant1 1082	. . . . . 6 |- ( ( ch /\ n = suc m /\ p = suc n ) -> f Fn n )
33	32	3ad2ant2 1083	. . . . 5 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ n = suc i ) ) -> f Fn n )
34		simp23 1096	. . . . 5 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ n = suc i ) ) -> p = suc n )
35	29, 33, 34, 13	bnj951 30846	. . . 4 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ n = suc i ) ) -> ( C e. _V /\ f Fn n /\ p = suc n /\ n = suc i ) )
36		bnj966.10	. . . . . . . . 9 |- D = ( om \ { (/) } )
37	36	bnj923 30838	. . . . . . . 8 |- ( n e. D -> n e. om )
38	30, 37	bnj769 30832	. . . . . . 7 |- ( ch -> n e. om )
39	38	3ad2ant1 1082	. . . . . 6 |- ( ( ch /\ n = suc m /\ p = suc n ) -> n e. om )
40		simp3 1063	. . . . . 6 |- ( ( i e. om /\ suc i e. p /\ n = suc i ) -> n = suc i )
41	39, 40	bnj240 30765	. . . . 5 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ n = suc i ) ) -> ( n e. om /\ n = suc i ) )
42		vex 3203	. . . . . . 7 |- i e. _V
43	42	bnj216 30800	. . . . . 6 |- ( n = suc i -> i e. n )
44	43	adantl 482	. . . . 5 |- ( ( n e. om /\ n = suc i ) -> i e. n )
45	41, 44	syl 17	. . . 4 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ n = suc i ) ) -> i e. n )
46		bnj658 30821	. . . . . . 7 |- ( ( C e. _V /\ f Fn n /\ p = suc n /\ n = suc i ) -> ( C e. _V /\ f Fn n /\ p = suc n ) )
47	46	anim1i 592	. . . . . 6 |- ( ( ( C e. _V /\ f Fn n /\ p = suc n /\ n = suc i ) /\ i e. n ) -> ( ( C e. _V /\ f Fn n /\ p = suc n ) /\ i e. n ) )
48		df-bnj17 30753	. . . . . 6 |- ( ( C e. _V /\ f Fn n /\ p = suc n /\ i e. n ) <-> ( ( C e. _V /\ f Fn n /\ p = suc n ) /\ i e. n ) )
49	47, 48	sylibr 224	. . . . 5 |- ( ( ( C e. _V /\ f Fn n /\ p = suc n /\ n = suc i ) /\ i e. n ) -> ( C e. _V /\ f Fn n /\ p = suc n /\ i e. n ) )
50	8	bnj945 30844	. . . . 5 |- ( ( C e. _V /\ f Fn n /\ p = suc n /\ i e. n ) -> ( G ` i ) = ( f ` i ) )
51	49, 50	syl 17	. . . 4 |- ( ( ( C e. _V /\ f Fn n /\ p = suc n /\ n = suc i ) /\ i e. n ) -> ( G ` i ) = ( f ` i ) )
52	35, 45, 51	syl2anc 693	. . 3 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ n = suc i ) ) -> ( G ` i ) = ( f ` i ) )
53	20, 8	bnj958 31010	. . . . 5 |- ( ( G ` i ) = ( f ` i ) -> A. y ( G ` i ) = ( f ` i ) )
54	53	bnj956 30847	. . . 4 |- ( ( G ` i ) = ( f ` i ) -> U_ y e. ( G ` i ) pred ( y , A , R ) = U_ y e. ( f ` i ) pred ( y , A , R ) )
55	54	eqeq2d 2632	. . 3 |- ( ( G ` i ) = ( f ` i ) -> ( ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) <-> ( G ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
56	52, 55	syl 17	. 2 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ n = suc i ) ) -> ( ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) <-> ( G ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
57	27, 56	mpbird 247	1 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ n = suc i ) ) -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   u. cun 3572   (/)c0 3915   {csn 4177   <.cop 4183   U_ciun 4520   suc csuc 5725   Fun wfun 5882   Fn wfn 5883   ` cfv 5888   omcom 7065   /\ w-bnj17 30752   predc-bnj14 30754   FrSe w-bnj15 30758

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  ax-reg 8497

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-id 5024  df-eprel 5029  df-fr 5073  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-bnj17 30753

This theorem is referenced by:  bnj910  31018