Theorem bnj944 29578

Description: Technical lemma for bnj69 29648. 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 458	. . . 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 29391	. . . . . . . 8 |- ( ( n e. D /\ f Fn n /\ ph /\ ps ) -> ( f Fn n /\ ph /\ ps ) )
4	2, 3	sylbi 198	. . . . . . 7 |- ( ch -> ( f Fn n /\ ph /\ ps ) )
5		bnj944.14	. . . . . . 7 |- ( ta <-> ( f Fn n /\ ph /\ ps ) )
6	4, 5	sylibr 215	. . . . . 6 |- ( ch -> ta )
7	6	3ad2ant1 1026	. . . . 5 |- ( ( ch /\ n = suc m /\ p = suc n ) -> ta )
8	7	adantl 467	. . . 4 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> ta )
9	2	bnj1232 29444	. . . . . . 7 |- ( ch -> n e. D )
10		vex 3081	. . . . . . . 8 |- m e. _V
11	10	bnj216 29369	. . . . . . 7 |- ( n = suc m -> m e. n )
12		id 23	. . . . . . 7 |- ( p = suc n -> p = suc n )
13	9, 11, 12	3anim123i 1190	. . . . . 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 991	. . . . . . 7 |- ( ( n e. D /\ p = suc n /\ m e. n ) <-> ( n e. D /\ m e. n /\ p = suc n ) )
16	14, 15	bitri 252	. . . . . 6 |- ( si <-> ( n e. D /\ m e. n /\ p = suc n ) )
17	13, 16	sylibr 215	. . . . 5 |- ( ( ch /\ n = suc m /\ p = suc n ) -> si )
18	17	adantl 467	. . . 4 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> si )
19		bnj253 29338	. . . 4 |- ( ( R FrSe A /\ X e. A /\ ta /\ si ) <-> ( ( R FrSe A /\ X e. A ) /\ ta /\ si ) )
20	1, 8, 18, 19	syl3anbrc 1189	. . 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 29577	. . . 4 |- ( ( R FrSe A /\ X e. A /\ ta /\ si ) -> U_ y e. ( f ` m ) pred ( y , A , R ) e. _V )
26	21, 25	syl5eqel 2512	. . 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 29390	. . . . . 6 |- ( ( n e. D /\ f Fn n /\ ph /\ ps ) -> ( n e. D /\ f Fn n /\ ph ) )
29	2, 28	sylbi 198	. . . . 5 |- ( ch -> ( n e. D /\ f Fn n /\ ph ) )
30	29	3ad2ant1 1026	. . . 4 |- ( ( ch /\ n = suc m /\ p = suc n ) -> ( n e. D /\ f Fn n /\ ph ) )
31		simp3 1007	. . . 4 |- ( ( ch /\ n = suc m /\ p = suc n ) -> p = suc n )
32		bnj291 29345	. . . 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 668	. . 3 |- ( ( ch /\ n = suc m /\ p = suc n ) -> ( n e. D /\ p = suc n /\ f Fn n /\ ph ) )
34	33	adantl 467	. 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 4182	. . . . . . . . 9 |- ( C = if ( C e. _V , C , (/) ) -> <. n , C >. = <. n , if ( C e. _V , C , (/) ) >. )
38	37	sneqd 4005	. . . . . . . 8 |- ( C = if ( C e. _V , C , (/) ) -> { <. n , C >. } = { <. n , if ( C e. _V , C , (/) ) >. } )
39	38	uneq2d 3617	. . . . . . 7 |- ( C = if ( C e. _V , C , (/) ) -> ( f u. { <. n , C >. } ) = ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) )
40	36, 39	syl5eq 2473	. . . . . 6 |- ( C = if ( C e. _V , C , (/) ) -> G = ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) )
41	40	sbceq1d 3301	. . . . 5 |- ( C = if ( C e. _V , C , (/) ) -> ( [. G / f ]. ph' <-> [. ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) / f ]. ph' ) )
42	35, 41	syl5bb 260	. . . 4 |- ( C = if ( C e. _V , C , (/) ) -> ( ph" <-> [. ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) / f ]. ph' ) )
43	42	imbi2d 317	. . 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 239	. . . 4 |- ( [. ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) / f ]. ph' <-> [. ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) / f ]. ph' )
46		eqid 2420	. . . 4 |- ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) = ( f u. { <. n , if ( C e. _V , C , (/) ) >. } )
47		0ex 4548	. . . . 5 |- (/) e. _V
48	47	elimel 3968	. . . 4 |- if ( C e. _V , C , (/) ) e. _V
49	23, 44, 45, 22, 46, 48	bnj929 29576	. . 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 3957	. 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 187   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1867   A.wral 2773   _Vcvv 3078   [.wsbc 3296   \ cdif 3430   u. cun 3431   (/)c0 3758   ifcif 3906   {csn 3993   <.cop 3999   U_ciun 4293   suc csuc 5435   Fn wfn 5587   ` cfv 5592   omcom 6697   /\ w-bnj17 29320   predc-bnj14 29322   FrSe w-bnj15 29326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pr 4652  ax-un 6588  ax-reg 8098

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-om 6698  df-bnj17 29321  df-bnj14 29323  df-bnj13 29325  df-bnj15 29327

This theorem is referenced by:  bnj910  29588