Theorem bnj944 29742

Description: Technical lemma for bnj69 29812. 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 459	. . . 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 29555	. . . . . . . 8 |- ( ( n e. D /\ f Fn n /\ ph /\ ps ) -> ( f Fn n /\ ph /\ ps ) )
4	2, 3	sylbi 199	. . . . . . 7 |- ( ch -> ( f Fn n /\ ph /\ ps ) )
5		bnj944.14	. . . . . . 7 |- ( ta <-> ( f Fn n /\ ph /\ ps ) )
6	4, 5	sylibr 216	. . . . . 6 |- ( ch -> ta )
7	6	3ad2ant1 1028	. . . . 5 |- ( ( ch /\ n = suc m /\ p = suc n ) -> ta )
8	7	adantl 468	. . . 4 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> ta )
9	2	bnj1232 29608	. . . . . . 7 |- ( ch -> n e. D )
10		vex 3047	. . . . . . . 8 |- m e. _V
11	10	bnj216 29533	. . . . . . 7 |- ( n = suc m -> m e. n )
12		id 22	. . . . . . 7 |- ( p = suc n -> p = suc n )
13	9, 11, 12	3anim123i 1192	. . . . . 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 993	. . . . . . 7 |- ( ( n e. D /\ p = suc n /\ m e. n ) <-> ( n e. D /\ m e. n /\ p = suc n ) )
16	14, 15	bitri 253	. . . . . 6 |- ( si <-> ( n e. D /\ m e. n /\ p = suc n ) )
17	13, 16	sylibr 216	. . . . 5 |- ( ( ch /\ n = suc m /\ p = suc n ) -> si )
18	17	adantl 468	. . . 4 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> si )
19		bnj253 29502	. . . 4 |- ( ( R FrSe A /\ X e. A /\ ta /\ si ) <-> ( ( R FrSe A /\ X e. A ) /\ ta /\ si ) )
20	1, 8, 18, 19	syl3anbrc 1191	. . 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 29741	. . . 4 |- ( ( R FrSe A /\ X e. A /\ ta /\ si ) -> U_ y e. ( f ` m ) pred ( y , A , R ) e. _V )
26	21, 25	syl5eqel 2532	. . 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 29554	. . . . . 6 |- ( ( n e. D /\ f Fn n /\ ph /\ ps ) -> ( n e. D /\ f Fn n /\ ph ) )
29	2, 28	sylbi 199	. . . . 5 |- ( ch -> ( n e. D /\ f Fn n /\ ph ) )
30	29	3ad2ant1 1028	. . . 4 |- ( ( ch /\ n = suc m /\ p = suc n ) -> ( n e. D /\ f Fn n /\ ph ) )
31		simp3 1009	. . . 4 |- ( ( ch /\ n = suc m /\ p = suc n ) -> p = suc n )
32		bnj291 29509	. . . 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 669	. . 3 |- ( ( ch /\ n = suc m /\ p = suc n ) -> ( n e. D /\ p = suc n /\ f Fn n /\ ph ) )
34	33	adantl 468	. 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 4166	. . . . . . . . 9 |- ( C = if ( C e. _V , C , (/) ) -> <. n , C >. = <. n , if ( C e. _V , C , (/) ) >. )
38	37	sneqd 3979	. . . . . . . 8 |- ( C = if ( C e. _V , C , (/) ) -> { <. n , C >. } = { <. n , if ( C e. _V , C , (/) ) >. } )
39	38	uneq2d 3587	. . . . . . 7 |- ( C = if ( C e. _V , C , (/) ) -> ( f u. { <. n , C >. } ) = ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) )
40	36, 39	syl5eq 2496	. . . . . 6 |- ( C = if ( C e. _V , C , (/) ) -> G = ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) )
41	40	sbceq1d 3271	. . . . 5 |- ( C = if ( C e. _V , C , (/) ) -> ( [. G / f ]. ph' <-> [. ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) / f ]. ph' ) )
42	35, 41	syl5bb 261	. . . 4 |- ( C = if ( C e. _V , C , (/) ) -> ( ph" <-> [. ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) / f ]. ph' ) )
43	42	imbi2d 318	. . 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 240	. . . 4 |- ( [. ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) / f ]. ph' <-> [. ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) / f ]. ph' )
46		eqid 2450	. . . 4 |- ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) = ( f u. { <. n , if ( C e. _V , C , (/) ) >. } )
47		0ex 4534	. . . . 5 |- (/) e. _V
48	47	elimel 3942	. . . 4 |- if ( C e. _V , C , (/) ) e. _V
49	23, 44, 45, 22, 46, 48	bnj929 29740	. . 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 3931	. 2 |- ( C e. _V -> ( ( n e. D /\ p = suc n /\ f Fn n /\ ph ) -> ph" ) )
51	27, 34, 50	sylc 62	1 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> ph" )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 984   = wceq 1443   e. wcel 1886   A.wral 2736   _Vcvv 3044   [.wsbc 3266   \ cdif 3400   u. cun 3401   (/)c0 3730   ifcif 3880   {csn 3967   <.cop 3973   U_ciun 4277   suc csuc 5424   Fn wfn 5576   ` cfv 5581   omcom 6689   /\ w-bnj17 29484   predc-bnj14 29486   FrSe w-bnj15 29490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pr 4638  ax-un 6580  ax-reg 8104

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-om 6690  df-bnj17 29485  df-bnj14 29487  df-bnj13 29489  df-bnj15 29491

This theorem is referenced by:  bnj910  29752