Theorem bnj944 33481

Description: Technical lemma for bnj69 33551. 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 457	. . . 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 33294	. . . . . . . 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 1017	. . . . 5 |- ( ( ch /\ n = suc m /\ p = suc n ) -> ta )
8	7	adantl 466	. . . 4 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> ta )
9	2	bnj1232 33347	. . . . . . 7 |- ( ch -> n e. D )
10		vex 3121	. . . . . . . 8 |- m e. _V
11	10	bnj216 33273	. . . . . . 7 |- ( n = suc m -> m e. n )
12		id 22	. . . . . . 7 |- ( p = suc n -> p = suc n )
13	9, 11, 12	3anim123i 1181	. . . . . 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 982	. . . . . . 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 466	. . . 4 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> si )
19		bnj253 33242	. . . 4 |- ( ( R FrSe A /\ X e. A /\ ta /\ si ) <-> ( ( R FrSe A /\ X e. A ) /\ ta /\ si ) )
20	1, 8, 18, 19	syl3anbrc 1180	. . 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 33480	. . . 4 |- ( ( R FrSe A /\ X e. A /\ ta /\ si ) -> U_ y e. ( f ` m ) pred ( y , A , R ) e. _V )
26	21, 25	syl5eqel 2559	. . 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 33293	. . . . . 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 1017	. . . 4 |- ( ( ch /\ n = suc m /\ p = suc n ) -> ( n e. D /\ f Fn n /\ ph ) )
31		simp3 998	. . . 4 |- ( ( ch /\ n = suc m /\ p = suc n ) -> p = suc n )
32		bnj291 33249	. . . 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 664	. . 3 |- ( ( ch /\ n = suc m /\ p = suc n ) -> ( n e. D /\ p = suc n /\ f Fn n /\ ph ) )
34	33	adantl 466	. 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 4220	. . . . . . . . 9 |- ( C = if ( C e. _V , C , (/) ) -> <. n , C >. = <. n , if ( C e. _V , C , (/) ) >. )
38	37	sneqd 4045	. . . . . . . 8 |- ( C = if ( C e. _V , C , (/) ) -> { <. n , C >. } = { <. n , if ( C e. _V , C , (/) ) >. } )
39	38	uneq2d 3663	. . . . . . 7 |- ( C = if ( C e. _V , C , (/) ) -> ( f u. { <. n , C >. } ) = ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) )
40	36, 39	syl5eq 2520	. . . . . 6 |- ( C = if ( C e. _V , C , (/) ) -> G = ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) )
41	40	sbceq1d 3341	. . . . 5 |- ( C = if ( C e. _V , C , (/) ) -> ( [. G / f ]. ph' <-> [. ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) / f ]. ph' ) )
42	35, 41	syl5bb 257	. . . 4 |- ( C = if ( C e. _V , C , (/) ) -> ( ph" <-> [. ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) / f ]. ph' ) )
43	42	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' ) ) )
44		bnj944.4	. . . 4 |- ( ph' <-> [. p / n ]. ph )
45		biid 236	. . . 4 |- ( [. ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) / f ]. ph' <-> [. ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) / f ]. ph' )
46		eqid 2467	. . . 4 |- ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) = ( f u. { <. n , if ( C e. _V , C , (/) ) >. } )
47		0ex 4583	. . . . 5 |- (/) e. _V
48	47	elimel 4008	. . . 4 |- if ( C e. _V , C , (/) ) e. _V
49	23, 44, 45, 22, 46, 48	bnj929 33479	. . 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 3997	. 2 |- ( C e. _V -> ( ( n e. D /\ p = suc n /\ f Fn n /\ ph ) -> ph" ) )
51	27, 34, 50	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 973   = wceq 1379   e. wcel 1767   A.wral 2817   _Vcvv 3118   [.wsbc 3336   \ cdif 3478   u. cun 3479   (/)c0 3790   ifcif 3945   {csn 4033   <.cop 4039   U_ciun 4331   suc csuc 4886   Fn wfn 5589   ` cfv 5594   omcom 6695   /\ w-bnj17 33224   predc-bnj14 33226   FrSe w-bnj15 33230

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pr 4692  ax-un 6587  ax-reg 8030

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6696  df-bnj17 33225  df-bnj14 33227  df-bnj13 33229  df-bnj15 33231

This theorem is referenced by:  bnj910  33491