Theorem bnj579 30984

Description: Technical lemma for bnj852 30991. 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
bnj579.1	|- ( ph <-> ( f ` (/) ) = pred ( x , A , R ) )
bnj579.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj579.3	|- D = ( om \ { (/) } )

Assertion

Ref	Expression
bnj579	|- ( n e. D -> E* f ( f Fn n /\ ph /\ ps ) )

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

Allowed substitution hints:   ph( x, y, f, i, n)   ps( x, y, f, i, n)   A( x, y, n)   D( x, y, i, n)   R( x, y, n)

Proof of Theorem bnj579

Dummy variables k g j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj579.1	. 2 |- ( ph <-> ( f ` (/) ) = pred ( x , A , R ) )
2		bnj579.2	. 2 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
3		biid 251	. 2 |- ( ( f Fn n /\ ph /\ ps ) <-> ( f Fn n /\ ph /\ ps ) )
4		biid 251	. 2 |- ( [. g / f ]. ph <-> [. g / f ]. ph )
5		biid 251	. 2 |- ( [. g / f ]. ps <-> [. g / f ]. ps )
6		biid 251	. 2 |- ( [. g / f ]. ( f Fn n /\ ph /\ ps ) <-> [. g / f ]. ( f Fn n /\ ph /\ ps ) )
7		bnj579.3	. 2 |- D = ( om \ { (/) } )
8		biid 251	. 2 |- ( ( ( n e. D /\ ( f Fn n /\ ph /\ ps ) /\ [. g / f ]. ( f Fn n /\ ph /\ ps ) ) -> ( f ` j ) = ( g ` j ) ) <-> ( ( n e. D /\ ( f Fn n /\ ph /\ ps ) /\ [. g / f ]. ( f Fn n /\ ph /\ ps ) ) -> ( f ` j ) = ( g ` j ) ) )
9		biid 251	. 2 |- ( A. k e. n ( k _E j -> [. k / j ]. ( ( n e. D /\ ( f Fn n /\ ph /\ ps ) /\ [. g / f ]. ( f Fn n /\ ph /\ ps ) ) -> ( f ` j ) = ( g ` j ) ) ) <-> A. k e. n ( k _E j -> [. k / j ]. ( ( n e. D /\ ( f Fn n /\ ph /\ ps ) /\ [. g / f ]. ( f Fn n /\ ph /\ ps ) ) -> ( f ` j ) = ( g ` j ) ) ) )
10	1, 2, 3, 4, 5, 6, 7, 8, 9	bnj580 30983	1 |- ( n e. D -> E* f ( f Fn n /\ ph /\ ps ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   E*wmo 2471   A.wral 2912   [.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   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-pow 4843  ax-pr 4906  ax-un 6949

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-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-fv 5896  df-om 7066  df-bnj17 30753

This theorem is referenced by:  bnj600  30989