Theorem bnj852 30991

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
bnj852.1	|- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
bnj852.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj852.3	|- D = ( om \ { (/) } )

Assertion

Ref	Expression
bnj852	|- ( ( R FrSe A /\ X e. A ) -> A. n e. D E! f ( f Fn n /\ ph /\ ps ) )

Distinct variable groups:   A, f, i, n, y   D, f, i, n   R, f, i, n, y   f, X, n

Allowed substitution hints:   ph( y, f, i, n)   ps( y, f, i, n)   D( y)   X( y, i)

Proof of Theorem bnj852

Dummy variables x z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elisset 3215	. . . . . 6 |- ( X e. A -> E. x x = X )
2	1	adantl 482	. . . . 5 |- ( ( R FrSe A /\ X e. A ) -> E. x x = X )
3	2	ancri 575	. . . 4 |- ( ( R FrSe A /\ X e. A ) -> ( E. x x = X /\ ( R FrSe A /\ X e. A ) ) )
4	3	bnj534 30808	. . 3 |- ( ( R FrSe A /\ X e. A ) -> E. x ( x = X /\ ( R FrSe A /\ X e. A ) ) )
5		eleq1 2689	. . . . . . 7 |- ( x = X -> ( x e. A <-> X e. A ) )
6	5	anbi2d 740	. . . . . 6 |- ( x = X -> ( ( R FrSe A /\ x e. A ) <-> ( R FrSe A /\ X e. A ) ) )
7	6	biimpar 502	. . . . 5 |- ( ( x = X /\ ( R FrSe A /\ X e. A ) ) -> ( R FrSe A /\ x e. A ) )
8		biid 251	. . . . . . . 8 |- ( A. z e. D ( z _E n -> [. z / n ]. ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ( f ` (/) ) = pred ( x , A , R ) /\ ps ) ) ) <-> A. z e. D ( z _E n -> [. z / n ]. ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ( f ` (/) ) = pred ( x , A , R ) /\ ps ) ) ) )
9		bnj852.3	. . . . . . . . 9 |- D = ( om \ { (/) } )
10		omex 8540	. . . . . . . . . 10 |- om e. _V
11		difexg 4808	. . . . . . . . . 10 |- ( om e. _V -> ( om \ { (/) } ) e. _V )
12	10, 11	ax-mp 5	. . . . . . . . 9 |- ( om \ { (/) } ) e. _V
13	9, 12	eqeltri 2697	. . . . . . . 8 |- D e. _V
14		zfregfr 8509	. . . . . . . 8 |- _E Fr D
15	8, 13, 14	bnj157 30929	. . . . . . 7 |- ( A. n e. D ( A. z e. D ( z _E n -> [. z / n ]. ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ( f ` (/) ) = pred ( x , A , R ) /\ ps ) ) ) -> ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ( f ` (/) ) = pred ( x , A , R ) /\ ps ) ) ) -> A. n e. D ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ( f ` (/) ) = pred ( x , A , R ) /\ ps ) ) )
16		biid 251	. . . . . . . . . 10 |- ( ( f ` (/) ) = pred ( x , A , R ) <-> ( f ` (/) ) = pred ( x , A , R ) )
17		bnj852.2	. . . . . . . . . 10 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
18		biid 251	. . . . . . . . . 10 |- ( ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ( f ` (/) ) = pred ( x , A , R ) /\ ps ) ) <-> ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ( f ` (/) ) = pred ( x , A , R ) /\ ps ) ) )
19	16, 17, 9, 18, 8	bnj153 30950	. . . . . . . . 9 |- ( n = 1o -> ( ( n e. D /\ A. z e. D ( z _E n -> [. z / n ]. ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ( f ` (/) ) = pred ( x , A , R ) /\ ps ) ) ) ) -> ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ( f ` (/) ) = pred ( x , A , R ) /\ ps ) ) ) )
20	16, 17, 9, 18, 8	bnj601 30990	. . . . . . . . 9 |- ( n =/= 1o -> ( ( n e. D /\ A. z e. D ( z _E n -> [. z / n ]. ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ( f ` (/) ) = pred ( x , A , R ) /\ ps ) ) ) ) -> ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ( f ` (/) ) = pred ( x , A , R ) /\ ps ) ) ) )
21	19, 20	pm2.61ine 2877	. . . . . . . 8 |- ( ( n e. D /\ A. z e. D ( z _E n -> [. z / n ]. ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ( f ` (/) ) = pred ( x , A , R ) /\ ps ) ) ) ) -> ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ( f ` (/) ) = pred ( x , A , R ) /\ ps ) ) )
22	21	ex 450	. . . . . . 7 |- ( n e. D -> ( A. z e. D ( z _E n -> [. z / n ]. ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ( f ` (/) ) = pred ( x , A , R ) /\ ps ) ) ) -> ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ( f ` (/) ) = pred ( x , A , R ) /\ ps ) ) ) )
23	15, 22	mprg 2926	. . . . . 6 |- A. n e. D ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ( f ` (/) ) = pred ( x , A , R ) /\ ps ) )
24		r19.21v 2960	. . . . . 6 |- ( A. n e. D ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ( f ` (/) ) = pred ( x , A , R ) /\ ps ) ) <-> ( ( R FrSe A /\ x e. A ) -> A. n e. D E! f ( f Fn n /\ ( f ` (/) ) = pred ( x , A , R ) /\ ps ) ) )
25	23, 24	mpbi 220	. . . . 5 |- ( ( R FrSe A /\ x e. A ) -> A. n e. D E! f ( f Fn n /\ ( f ` (/) ) = pred ( x , A , R ) /\ ps ) )
26	7, 25	syl 17	. . . 4 |- ( ( x = X /\ ( R FrSe A /\ X e. A ) ) -> A. n e. D E! f ( f Fn n /\ ( f ` (/) ) = pred ( x , A , R ) /\ ps ) )
27		bnj602 30985	. . . . . . . . . 10 |- ( x = X -> pred ( x , A , R ) = pred ( X , A , R ) )
28	27	eqeq2d 2632	. . . . . . . . 9 |- ( x = X -> ( ( f ` (/) ) = pred ( x , A , R ) <-> ( f ` (/) ) = pred ( X , A , R ) ) )
29		bnj852.1	. . . . . . . . 9 |- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
30	28, 29	syl6bbr 278	. . . . . . . 8 |- ( x = X -> ( ( f ` (/) ) = pred ( x , A , R ) <-> ph ) )
31	30	3anbi2d 1404	. . . . . . 7 |- ( x = X -> ( ( f Fn n /\ ( f ` (/) ) = pred ( x , A , R ) /\ ps ) <-> ( f Fn n /\ ph /\ ps ) ) )
32	31	eubidv 2490	. . . . . 6 |- ( x = X -> ( E! f ( f Fn n /\ ( f ` (/) ) = pred ( x , A , R ) /\ ps ) <-> E! f ( f Fn n /\ ph /\ ps ) ) )
33	32	ralbidv 2986	. . . . 5 |- ( x = X -> ( A. n e. D E! f ( f Fn n /\ ( f ` (/) ) = pred ( x , A , R ) /\ ps ) <-> A. n e. D E! f ( f Fn n /\ ph /\ ps ) ) )
34	33	adantr 481	. . . 4 |- ( ( x = X /\ ( R FrSe A /\ X e. A ) ) -> ( A. n e. D E! f ( f Fn n /\ ( f ` (/) ) = pred ( x , A , R ) /\ ps ) <-> A. n e. D E! f ( f Fn n /\ ph /\ ps ) ) )
35	26, 34	mpbid 222	. . 3 |- ( ( x = X /\ ( R FrSe A /\ X e. A ) ) -> A. n e. D E! f ( f Fn n /\ ph /\ ps ) )
36	4, 35	bnj593 30815	. 2 |- ( ( R FrSe A /\ X e. A ) -> E. x A. n e. D E! f ( f Fn n /\ ph /\ ps ) )
37	36	bnj937 30842	1 |- ( ( R FrSe A /\ X e. A ) -> A. n e. D E! f ( f Fn n /\ ph /\ ps ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   E!weu 2470   A.wral 2912   _Vcvv 3200   [.wsbc 3435   \ cdif 3571   (/)c0 3915   {csn 4177   U_ciun 4520   class class class wbr 4653   _E cep 5028   suc csuc 5725   Fn wfn 5883   ` cfv 5888   omcom 7065   1oc1o 7553   predc-bnj14 30754   FrSe w-bnj15 30758

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  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-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-bnj17 30753  df-bnj14 30755  df-bnj13 30757  df-bnj15 30759

This theorem is referenced by:  bnj864  30992  bnj865  30993  bnj906  31000