Theorem bnj944 31765

Description: Technical lemma for bnj69 31835. 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 454	. . . 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 31578	. . . . . . . 8 |- ( ( n e. D /\ f Fn n /\ ph /\ ps ) -> ( f Fn n /\ ph /\ ps ) )
4	2, 3	sylbi 195	. . . . . . 7 |- ( ch -> ( f Fn n /\ ph /\ ps ) )
5		bnj944.14	. . . . . . 7 |- ( ta <-> ( f Fn n /\ ph /\ ps ) )
6	4, 5	sylibr 212	. . . . . 6 |- ( ch -> ta )
7	6	3ad2ant1 1004	. . . . 5 |- ( ( ch /\ n = suc m /\ p = suc n ) -> ta )
8	7	adantl 463	. . . 4 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> ta )
9	2	bnj1232 31631	. . . . . . 7 |- ( ch -> n e. D )
10		vex 2973	. . . . . . . 8 |- m e. _V
11	10	bnj216 31557	. . . . . . 7 |- ( n = suc m -> m e. n )
12		id 22	. . . . . . 7 |- ( p = suc n -> p = suc n )
13	9, 11, 12	3anim123i 1168	. . . . . 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 969	. . . . . . 7 |- ( ( n e. D /\ p = suc n /\ m e. n ) <-> ( n e. D /\ m e. n /\ p = suc n ) )
16	14, 15	bitri 249	. . . . . 6 |- ( si <-> ( n e. D /\ m e. n /\ p = suc n ) )
17	13, 16	sylibr 212	. . . . 5 |- ( ( ch /\ n = suc m /\ p = suc n ) -> si )
18	17	adantl 463	. . . 4 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> si )
19		bnj253 31526	. . . 4 |- ( ( R FrSe A /\ X e. A /\ ta /\ si ) <-> ( ( R FrSe A /\ X e. A ) /\ ta /\ si ) )
20	1, 8, 18, 19	syl3anbrc 1167	. . 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 31764	. . . 4 |- ( ( R FrSe A /\ X e. A /\ ta /\ si ) -> U_ y e. ( f ` m ) pred ( y , A , R ) e. _V )
26	21, 25	syl5eqel 2525	. . 3 |- ( ( R FrSe A /\ X e. A /\ ta /\ si ) -> C e. _V )
27	20, 26	syl 16	. 2 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> C e. _V )
28		bnj658 31577	. . . . . 6 |- ( ( n e. D /\ f Fn n /\ ph /\ ps ) -> ( n e. D /\ f Fn n /\ ph ) )
29	2, 28	sylbi 195	. . . . 5 |- ( ch -> ( n e. D /\ f Fn n /\ ph ) )
30	29	3ad2ant1 1004	. . . 4 |- ( ( ch /\ n = suc m /\ p = suc n ) -> ( n e. D /\ f Fn n /\ ph ) )
31		simp3 985	. . . 4 |- ( ( ch /\ n = suc m /\ p = suc n ) -> p = suc n )
32		bnj291 31533	. . . 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 659	. . 3 |- ( ( ch /\ n = suc m /\ p = suc n ) -> ( n e. D /\ p = suc n /\ f Fn n /\ ph ) )
34	33	adantl 463	. 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 4057	. . . . . . . . 9 |- ( C = if ( C e. _V , C , (/) ) -> <. n , C >. = <. n , if ( C e. _V , C , (/) ) >. )
38	37	sneqd 3886	. . . . . . . 8 |- ( C = if ( C e. _V , C , (/) ) -> { <. n , C >. } = { <. n , if ( C e. _V , C , (/) ) >. } )
39	38	uneq2d 3507	. . . . . . 7 |- ( C = if ( C e. _V , C , (/) ) -> ( f u. { <. n , C >. } ) = ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) )
40	36, 39	syl5eq 2485	. . . . . 6 |- ( C = if ( C e. _V , C , (/) ) -> G = ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) )
41		dfsbcq 3185	. . . . . 6 |- ( G = ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) -> ( [. G / f ]. ph' <-> [. ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) / f ]. ph' ) )
42	40, 41	syl 16	. . . . 5 |- ( C = if ( C e. _V , C , (/) ) -> ( [. G / f ]. ph' <-> [. ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) / f ]. ph' ) )
43	35, 42	syl5bb 257	. . . 4 |- ( C = if ( C e. _V , C , (/) ) -> ( ph" <-> [. ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) / f ]. ph' ) )
44	43	imbi2d 316	. . 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' ) ) )
45		bnj944.4	. . . 4 |- ( ph' <-> [. p / n ]. ph )
46		biid 236	. . . 4 |- ( [. ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) / f ]. ph' <-> [. ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) / f ]. ph' )
47		eqid 2441	. . . 4 |- ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) = ( f u. { <. n , if ( C e. _V , C , (/) ) >. } )
48		0ex 4419	. . . . 5 |- (/) e. _V
49	48	elimel 3849	. . . 4 |- if ( C e. _V , C , (/) ) e. _V
50	23, 45, 46, 22, 47, 49	bnj929 31763	. . 3 |- ( ( n e. D /\ p = suc n /\ f Fn n /\ ph ) -> [. ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) / f ]. ph' )
51	44, 50	dedth 3838	. 2 |- ( C e. _V -> ( ( n e. D /\ p = suc n /\ f Fn n /\ ph ) -> ph" ) )
52	27, 34, 51	sylc 60	1 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> ph" )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 960   = wceq 1364   e. wcel 1761   A.wral 2713   _Vcvv 2970   [.wsbc 3183   \ cdif 3322   u. cun 3323   (/)c0 3634   ifcif 3788   {csn 3874   <.cop 3880   U_ciun 4168   suc csuc 4717   Fn wfn 5410   ` cfv 5415   omcom 6475   /\ w-bnj17 31508   predc-bnj14 31510   FrSe w-bnj15 31514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pr 4528  ax-un 6371  ax-reg 7803

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-om 6476  df-bnj17 31509  df-bnj14 31511  df-bnj13 31513  df-bnj15 31515

This theorem is referenced by:  bnj910  31775