Theorem bnj944 31008

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
bnj944.1	|- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
bnj944.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj944.3	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj944.4	|- ( ph' <-> [. p / n ]. ph )
bnj944.7	|- ( ph" <-> [. G / f ]. ph' )
bnj944.10	|- D = ( om \ { (/) } )
bnj944.12	|- C = U_ y e. ( f ` m ) pred ( y , A , R )
bnj944.13	|- G = ( f u. { <. n , C >. } )
bnj944.14	|- ( ta <-> ( f Fn n /\ ph /\ ps ) )
bnj944.15	|- ( si <-> ( n e. D /\ p = suc n /\ m e. n ) )

Assertion

Ref	Expression
bnj944	|- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> ph" )

Distinct variable groups:   A, f, i, m, n   y, A, f, i, m   R, f, i, m, n   y, R   f, X, 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)   ta( y, f, i, m, n, p)   si( y, f, i, m, n, p)   A( p)   C( y, f, i, m, n, p)   D( y, f, i, m, n, p)   R( p)   G( y, f, i, m, n, p)   X( y, i, m, p)   ph'( y, f, i, m, n, p)   ph"( y, f, i, m, n, p)

Proof of Theorem bnj944

Step	Hyp	Ref	Expression
1		simpl 473	. . . 4 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> ( R FrSe A /\ X e. A ) )
2		bnj944.3	. . . . . . . 8 |- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
3		bnj667 30822	. . . . . . . 8 |- ( ( n e. D /\ f Fn n /\ ph /\ ps ) -> ( f Fn n /\ ph /\ ps ) )
4	2, 3	sylbi 207	. . . . . . 7 |- ( ch -> ( f Fn n /\ ph /\ ps ) )
5		bnj944.14	. . . . . . 7 |- ( ta <-> ( f Fn n /\ ph /\ ps ) )
6	4, 5	sylibr 224	. . . . . 6 |- ( ch -> ta )
7	6	3ad2ant1 1082	. . . . 5 |- ( ( ch /\ n = suc m /\ p = suc n ) -> ta )
8	7	adantl 482	. . . 4 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> ta )
9	2	bnj1232 30874	. . . . . . 7 |- ( ch -> n e. D )
10		vex 3203	. . . . . . . 8 |- m e. _V
11	10	bnj216 30800	. . . . . . 7 |- ( n = suc m -> m e. n )
12		id 22	. . . . . . 7 |- ( p = suc n -> p = suc n )
13	9, 11, 12	3anim123i 1247	. . . . . 6 |- ( ( ch /\ n = suc m /\ p = suc n ) -> ( n e. D /\ m e. n /\ p = suc n ) )
14		bnj944.15	. . . . . . 7 |- ( si <-> ( n e. D /\ p = suc n /\ m e. n ) )
15		3ancomb 1047	. . . . . . 7 |- ( ( n e. D /\ p = suc n /\ m e. n ) <-> ( n e. D /\ m e. n /\ p = suc n ) )
16	14, 15	bitri 264	. . . . . 6 |- ( si <-> ( n e. D /\ m e. n /\ p = suc n ) )
17	13, 16	sylibr 224	. . . . 5 |- ( ( ch /\ n = suc m /\ p = suc n ) -> si )
18	17	adantl 482	. . . 4 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> si )
19		bnj253 30770	. . . 4 |- ( ( R FrSe A /\ X e. A /\ ta /\ si ) <-> ( ( R FrSe A /\ X e. A ) /\ ta /\ si ) )
20	1, 8, 18, 19	syl3anbrc 1246	. . 3 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> ( R FrSe A /\ X e. A /\ ta /\ si ) )
21		bnj944.12	. . . 4 |- C = U_ y e. ( f ` m ) pred ( y , A , R )
22		bnj944.10	. . . . 5 |- D = ( om \ { (/) } )
23		bnj944.1	. . . . 5 |- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
24		bnj944.2	. . . . 5 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
25	22, 5, 14, 23, 24	bnj938 31007	. . . 4 |- ( ( R FrSe A /\ X e. A /\ ta /\ si ) -> U_ y e. ( f ` m ) pred ( y , A , R ) e. _V )
26	21, 25	syl5eqel 2705	. . 3 |- ( ( R FrSe A /\ X e. A /\ ta /\ si ) -> C e. _V )
27	20, 26	syl 17	. 2 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> C e. _V )
28		bnj658 30821	. . . . . 6 |- ( ( n e. D /\ f Fn n /\ ph /\ ps ) -> ( n e. D /\ f Fn n /\ ph ) )
29	2, 28	sylbi 207	. . . . 5 |- ( ch -> ( n e. D /\ f Fn n /\ ph ) )
30	29	3ad2ant1 1082	. . . 4 |- ( ( ch /\ n = suc m /\ p = suc n ) -> ( n e. D /\ f Fn n /\ ph ) )
31		simp3 1063	. . . 4 |- ( ( ch /\ n = suc m /\ p = suc n ) -> p = suc n )
32		bnj291 30777	. . . 4 |- ( ( n e. D /\ p = suc n /\ f Fn n /\ ph ) <-> ( ( n e. D /\ f Fn n /\ ph ) /\ p = suc n ) )
33	30, 31, 32	sylanbrc 698	. . 3 |- ( ( ch /\ n = suc m /\ p = suc n ) -> ( n e. D /\ p = suc n /\ f Fn n /\ ph ) )
34	33	adantl 482	. 2 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> ( n e. D /\ p = suc n /\ f Fn n /\ ph ) )
35		bnj944.7	. . . . 5 |- ( ph" <-> [. G / f ]. ph' )
36		bnj944.13	. . . . . . 7 |- G = ( f u. { <. n , C >. } )
37		opeq2 4403	. . . . . . . . 9 |- ( C = if ( C e. _V , C , (/) ) -> <. n , C >. = <. n , if ( C e. _V , C , (/) ) >. )
38	37	sneqd 4189	. . . . . . . 8 |- ( C = if ( C e. _V , C , (/) ) -> { <. n , C >. } = { <. n , if ( C e. _V , C , (/) ) >. } )
39	38	uneq2d 3767	. . . . . . 7 |- ( C = if ( C e. _V , C , (/) ) -> ( f u. { <. n , C >. } ) = ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) )
40	36, 39	syl5eq 2668	. . . . . 6 |- ( C = if ( C e. _V , C , (/) ) -> G = ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) )
41	40	sbceq1d 3440	. . . . 5 |- ( C = if ( C e. _V , C , (/) ) -> ( [. G / f ]. ph' <-> [. ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) / f ]. ph' ) )
42	35, 41	syl5bb 272	. . . 4 |- ( C = if ( C e. _V , C , (/) ) -> ( ph" <-> [. ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) / f ]. ph' ) )
43	42	imbi2d 330	. . 3 |- ( C = if ( C e. _V , C , (/) ) -> ( ( ( n e. D /\ p = suc n /\ f Fn n /\ ph ) -> ph" ) <-> ( ( n e. D /\ p = suc n /\ f Fn n /\ ph ) -> [. ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) / f ]. ph' ) ) )
44		bnj944.4	. . . 4 |- ( ph' <-> [. p / n ]. ph )
45		biid 251	. . . 4 |- ( [. ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) / f ]. ph' <-> [. ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) / f ]. ph' )
46		eqid 2622	. . . 4 |- ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) = ( f u. { <. n , if ( C e. _V , C , (/) ) >. } )
47		0ex 4790	. . . . 5 |- (/) e. _V
48	47	elimel 4150	. . . 4 |- if ( C e. _V , C , (/) ) e. _V
49	23, 44, 45, 22, 46, 48	bnj929 31006	. . 3 |- ( ( n e. D /\ p = suc n /\ f Fn n /\ ph ) -> [. ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) / f ]. ph' )
50	43, 49	dedth 4139	. 2 |- ( C e. _V -> ( ( n e. D /\ p = suc n /\ f Fn n /\ ph ) -> ph" ) )
51	27, 34, 50	sylc 65	1 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> ph" )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   [.wsbc 3435   \ cdif 3571   u. cun 3572   (/)c0 3915   ifcif 4086   {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-rep 4771  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-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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-bnj17 30753  df-bnj14 30755  df-bnj13 30757  df-bnj15 30759

This theorem is referenced by:  bnj910  31018