Theorem bnj929 31006

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
bnj929.1	|- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
bnj929.4	|- ( ph' <-> [. p / n ]. ph )
bnj929.7	|- ( ph" <-> [. G / f ]. ph' )
bnj929.10	|- D = ( om \ { (/) } )
bnj929.13	|- G = ( f u. { <. n , C >. } )
bnj929.50	|- C e. _V

Assertion

Ref	Expression
bnj929	|- ( ( n e. D /\ p = suc n /\ f Fn n /\ ph ) -> ph" )

Distinct variable groups:   A, f, n   R, f, n   f, X, n

Allowed substitution hints:   ph( f, n, p)   A( p)   C( f, n, p)   D( f, n, p)   R( p)   G( f, n, p)   X( p)   ph'( f, n, p)   ph"( f, n, p)

Proof of Theorem bnj929

Step	Hyp	Ref	Expression
1		bnj645 30820	. 2 |- ( ( n e. D /\ p = suc n /\ f Fn n /\ ph ) -> ph )
2		bnj334 30779	. . . . . . 7 |- ( ( n e. D /\ p = suc n /\ f Fn n /\ ph ) <-> ( f Fn n /\ n e. D /\ p = suc n /\ ph ) )
3		bnj257 30773	. . . . . . 7 |- ( ( f Fn n /\ n e. D /\ p = suc n /\ ph ) <-> ( f Fn n /\ n e. D /\ ph /\ p = suc n ) )
4	2, 3	bitri 264	. . . . . 6 |- ( ( n e. D /\ p = suc n /\ f Fn n /\ ph ) <-> ( f Fn n /\ n e. D /\ ph /\ p = suc n ) )
5		bnj345 30780	. . . . . 6 |- ( ( f Fn n /\ n e. D /\ ph /\ p = suc n ) <-> ( p = suc n /\ f Fn n /\ n e. D /\ ph ) )
6		bnj253 30770	. . . . . 6 |- ( ( p = suc n /\ f Fn n /\ n e. D /\ ph ) <-> ( ( p = suc n /\ f Fn n ) /\ n e. D /\ ph ) )
7	4, 5, 6	3bitri 286	. . . . 5 |- ( ( n e. D /\ p = suc n /\ f Fn n /\ ph ) <-> ( ( p = suc n /\ f Fn n ) /\ n e. D /\ ph ) )
8	7	simp1bi 1076	. . . 4 |- ( ( n e. D /\ p = suc n /\ f Fn n /\ ph ) -> ( p = suc n /\ f Fn n ) )
9		bnj929.13	. . . . . 6 |- G = ( f u. { <. n , C >. } )
10		bnj929.50	. . . . . 6 |- C e. _V
11	9, 10	bnj927 30839	. . . . 5 |- ( ( p = suc n /\ f Fn n ) -> G Fn p )
12	11	bnj930 30840	. . . 4 |- ( ( p = suc n /\ f Fn n ) -> Fun G )
13	8, 12	syl 17	. . 3 |- ( ( n e. D /\ p = suc n /\ f Fn n /\ ph ) -> Fun G )
14	9	bnj931 30841	. . . 4 |- f C_ G
15	14	a1i 11	. . 3 |- ( ( n e. D /\ p = suc n /\ f Fn n /\ ph ) -> f C_ G )
16		bnj268 30775	. . . . . 6 |- ( ( n e. D /\ f Fn n /\ p = suc n /\ ph ) <-> ( n e. D /\ p = suc n /\ f Fn n /\ ph ) )
17		bnj253 30770	. . . . . 6 |- ( ( n e. D /\ f Fn n /\ p = suc n /\ ph ) <-> ( ( n e. D /\ f Fn n ) /\ p = suc n /\ ph ) )
18	16, 17	bitr3i 266	. . . . 5 |- ( ( n e. D /\ p = suc n /\ f Fn n /\ ph ) <-> ( ( n e. D /\ f Fn n ) /\ p = suc n /\ ph ) )
19	18	simp1bi 1076	. . . 4 |- ( ( n e. D /\ p = suc n /\ f Fn n /\ ph ) -> ( n e. D /\ f Fn n ) )
20		fndm 5990	. . . . 5 |- ( f Fn n -> dom f = n )
21		bnj929.10	. . . . . 6 |- D = ( om \ { (/) } )
22	21	bnj529 30811	. . . . 5 |- ( n e. D -> (/) e. n )
23		eleq2 2690	. . . . . 6 |- ( dom f = n -> ( (/) e. dom f <-> (/) e. n ) )
24	23	biimpar 502	. . . . 5 |- ( ( dom f = n /\ (/) e. n ) -> (/) e. dom f )
25	20, 22, 24	syl2anr 495	. . . 4 |- ( ( n e. D /\ f Fn n ) -> (/) e. dom f )
26	19, 25	syl 17	. . 3 |- ( ( n e. D /\ p = suc n /\ f Fn n /\ ph ) -> (/) e. dom f )
27	13, 15, 26	bnj1502 30918	. 2 |- ( ( n e. D /\ p = suc n /\ f Fn n /\ ph ) -> ( G ` (/) ) = ( f ` (/) ) )
28		bnj929.1	. . 3 |- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
29		bnj929.4	. . 3 |- ( ph' <-> [. p / n ]. ph )
30		bnj929.7	. . 3 |- ( ph" <-> [. G / f ]. ph' )
31	9	bnj918 30836	. . 3 |- G e. _V
32	28, 29, 30, 31	bnj934 31005	. 2 |- ( ( ph /\ ( G ` (/) ) = ( f ` (/) ) ) -> ph" )
33	1, 27, 32	syl2anc 693	1 |- ( ( n e. D /\ p = suc n /\ f Fn n /\ ph ) -> ph" )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   [.wsbc 3435   \ cdif 3571   u. cun 3572   C_ wss 3574   (/)c0 3915   {csn 4177   <.cop 4183   dom cdm 5114   suc csuc 5725   Fun wfun 5882   Fn wfn 5883   ` cfv 5888   omcom 7065   /\ w-bnj17 30752   predc-bnj14 30754

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-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-rab 2921  df-v 3202  df-sbc 3436  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-br 4654  df-opab 4713  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-res 5126  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-om 7066  df-bnj17 30753

This theorem is referenced by:  bnj944  31008