Theorem bnj964 31013

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
bnj964.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj964.3	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj964.5	|- ( ps' <-> [. p / n ]. ps )
bnj964.8	|- ( ps" <-> [. G / f ]. ps' )
bnj964.12	|- C = U_ y e. ( f ` m ) pred ( y , A , R )
bnj964.13	|- G = ( f u. { <. n , C >. } )
bnj964.96	|- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ suc i e. n ) ) -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) )
bnj964.165	|- ( ( ( 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 ) )

Assertion

Ref	Expression
bnj964	|- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> ps" )

Distinct variable groups:   A, f, i, n   D, i   i, G   R, f, i, n   i, X   f, p, i   y, f, i, n   i, m   ph, i

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

Proof of Theorem bnj964

Step	Hyp	Ref	Expression
1		nfv 1843	. . . 4 |- F/ i ( R FrSe A /\ X e. A )
2		bnj964.2	. . . . . . . 8 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
3	2	bnj1095 30852	. . . . . . 7 |- ( ps -> A. i ps )
4		bnj964.3	. . . . . . 7 |- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
5	3, 4	bnj1096 30853	. . . . . 6 |- ( ch -> A. i ch )
6	5	nf5i 2024	. . . . 5 |- F/ i ch
7		nfv 1843	. . . . 5 |- F/ i n = suc m
8		nfv 1843	. . . . 5 |- F/ i p = suc n
9	6, 7, 8	nf3an 1831	. . . 4 |- F/ i ( ch /\ n = suc m /\ p = suc n )
10	1, 9	nfan 1828	. . 3 |- F/ i ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) )
11		bnj255 30771	. . . . 5 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ i e. om /\ suc i e. p ) <-> ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p ) ) )
12		bnj645 30820	. . . . . . 7 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ i e. om /\ suc i e. p ) -> suc i e. p )
13		simp3 1063	. . . . . . . 8 |- ( ( ch /\ n = suc m /\ p = suc n ) -> p = suc n )
14	13	bnj706 30824	. . . . . . 7 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ i e. om /\ suc i e. p ) -> p = suc n )
15		eleq2 2690	. . . . . . . . 9 |- ( p = suc n -> ( suc i e. p <-> suc i e. suc n ) )
16	15	biimpac 503	. . . . . . . 8 |- ( ( suc i e. p /\ p = suc n ) -> suc i e. suc n )
17		elsuci 5791	. . . . . . . . 9 |- ( suc i e. suc n -> ( suc i e. n \/ suc i = n ) )
18		eqcom 2629	. . . . . . . . . 10 |- ( suc i = n <-> n = suc i )
19	18	orbi2i 541	. . . . . . . . 9 |- ( ( suc i e. n \/ suc i = n ) <-> ( suc i e. n \/ n = suc i ) )
20	17, 19	sylib 208	. . . . . . . 8 |- ( suc i e. suc n -> ( suc i e. n \/ n = suc i ) )
21	16, 20	syl 17	. . . . . . 7 |- ( ( suc i e. p /\ p = suc n ) -> ( suc i e. n \/ n = suc i ) )
22	12, 14, 21	syl2anc 693	. . . . . 6 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ i e. om /\ suc i e. p ) -> ( suc i e. n \/ n = suc i ) )
23		df-3an 1039	. . . . . . . . . . . . 13 |- ( ( i e. om /\ suc i e. p /\ suc i e. n ) <-> ( ( i e. om /\ suc i e. p ) /\ suc i e. n ) )
24	23	3anbi3i 1255	. . . . . . . . . . . 12 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ suc i e. n ) ) <-> ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( ( i e. om /\ suc i e. p ) /\ suc i e. n ) ) )
25		bnj255 30771	. . . . . . . . . . . 12 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p ) /\ suc i e. n ) <-> ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( ( i e. om /\ suc i e. p ) /\ suc i e. n ) ) )
26	24, 25	bitr4i 267	. . . . . . . . . . 11 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ suc i e. n ) ) <-> ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p ) /\ suc i e. n ) )
27		bnj345 30780	. . . . . . . . . . 11 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p ) /\ suc i e. n ) <-> ( suc i e. n /\ ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p ) ) )
28		bnj252 30769	. . . . . . . . . . 11 |- ( ( suc i e. n /\ ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p ) ) <-> ( suc i e. n /\ ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p ) ) ) )
29	26, 27, 28	3bitri 286	. . . . . . . . . 10 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ suc i e. n ) ) <-> ( suc i e. n /\ ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p ) ) ) )
30	11	anbi2i 730	. . . . . . . . . 10 |- ( ( suc i e. n /\ ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ i e. om /\ suc i e. p ) ) <-> ( suc i e. n /\ ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p ) ) ) )
31	29, 30	bitr4i 267	. . . . . . . . 9 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ suc i e. n ) ) <-> ( suc i e. n /\ ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ i e. om /\ suc i e. p ) ) )
32		bnj964.96	. . . . . . . . 9 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ suc i e. n ) ) -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) )
33	31, 32	sylbir 225	. . . . . . . 8 |- ( ( suc i e. n /\ ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ i e. om /\ suc i e. p ) ) -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) )
34	33	ex 450	. . . . . . 7 |- ( suc i e. n -> ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ i e. om /\ suc i e. p ) -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) ) )
35		df-3an 1039	. . . . . . . . . . . . 13 |- ( ( i e. om /\ suc i e. p /\ n = suc i ) <-> ( ( i e. om /\ suc i e. p ) /\ n = suc i ) )
36	35	3anbi3i 1255	. . . . . . . . . . . 12 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ n = suc i ) ) <-> ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( ( i e. om /\ suc i e. p ) /\ n = suc i ) ) )
37		bnj255 30771	. . . . . . . . . . . 12 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p ) /\ n = suc i ) <-> ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( ( i e. om /\ suc i e. p ) /\ n = suc i ) ) )
38	36, 37	bitr4i 267	. . . . . . . . . . 11 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ n = suc i ) ) <-> ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p ) /\ n = suc i ) )
39		bnj345 30780	. . . . . . . . . . 11 |- ( ( ( 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 /\ ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p ) ) )
40		bnj252 30769	. . . . . . . . . . 11 |- ( ( n = suc i /\ ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p ) ) <-> ( n = suc i /\ ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p ) ) ) )
41	38, 39, 40	3bitri 286	. . . . . . . . . 10 |- ( ( ( 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 /\ ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p ) ) ) )
42	11	anbi2i 730	. . . . . . . . . 10 |- ( ( n = suc i /\ ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ i e. om /\ suc i e. p ) ) <-> ( n = suc i /\ ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p ) ) ) )
43	41, 42	bitr4i 267	. . . . . . . . 9 |- ( ( ( 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 /\ ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ i e. om /\ suc i e. p ) ) )
44		bnj964.165	. . . . . . . . 9 |- ( ( ( 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 ) )
45	43, 44	sylbir 225	. . . . . . . 8 |- ( ( n = suc i /\ ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ i e. om /\ suc i e. p ) ) -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) )
46	45	ex 450	. . . . . . 7 |- ( n = suc i -> ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ i e. om /\ suc i e. p ) -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) ) )
47	34, 46	jaoi 394	. . . . . 6 |- ( ( suc i e. n \/ n = suc i ) -> ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ i e. om /\ suc i e. p ) -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) ) )
48	22, 47	mpcom 38	. . . . 5 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ i e. om /\ suc i e. p ) -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) )
49	11, 48	sylbir 225	. . . 4 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p ) ) -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) )
50	49	3expia 1267	. . 3 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> ( ( i e. om /\ suc i e. p ) -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) ) )
51	10, 50	alrimi 2082	. 2 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> A. i ( ( i e. om /\ suc i e. p ) -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) ) )
52		bnj964.5	. . . . 5 |- ( ps' <-> [. p / n ]. ps )
53		vex 3203	. . . . 5 |- p e. _V
54	2, 52, 53	bnj539 30961	. . . 4 |- ( ps' <-> A. i e. om ( suc i e. p -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
55		bnj964.8	. . . 4 |- ( ps" <-> [. G / f ]. ps' )
56		bnj964.12	. . . 4 |- C = U_ y e. ( f ` m ) pred ( y , A , R )
57		bnj964.13	. . . 4 |- G = ( f u. { <. n , C >. } )
58	54, 55, 56, 57	bnj965 31012	. . 3 |- ( ps" <-> A. i e. om ( suc i e. p -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) ) )
59	58	bnj115 30791	. 2 |- ( ps" <-> A. i ( ( i e. om /\ suc i e. p ) -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) ) )
60	51, 59	sylibr 224	1 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> ps" )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   A.wral 2912   [.wsbc 3435   u. cun 3572   {csn 4177   <.cop 4183   U_ciun 4520   suc csuc 5725   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

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  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-suc 5729  df-iota 5851  df-fv 5896  df-bnj17 30753

This theorem is referenced by:  bnj910  31018