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Theorem rdgsucmptnf 7096
Description: The value of the recursive definition generator at a successor (special case where the characteristic function is an ordered-pair class abstraction and where the mapping class  D is a proper class). This is a technical lemma that can be used together with rdgsucmptf 7095 to help eliminate redundant sethood antecedents. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
rdgsucmptf.1  |-  F/_ x A
rdgsucmptf.2  |-  F/_ x B
rdgsucmptf.3  |-  F/_ x D
rdgsucmptf.4  |-  F  =  rec ( ( x  e.  _V  |->  C ) ,  A )
rdgsucmptf.5  |-  ( x  =  ( F `  B )  ->  C  =  D )
Assertion
Ref Expression
rdgsucmptnf  |-  ( -.  D  e.  _V  ->  ( F `  suc  B
)  =  (/) )

Proof of Theorem rdgsucmptnf
StepHypRef Expression
1 rdgsucmptf.4 . . 3  |-  F  =  rec ( ( x  e.  _V  |->  C ) ,  A )
21fveq1i 5867 . 2  |-  ( F `
 suc  B )  =  ( rec (
( x  e.  _V  |->  C ) ,  A
) `  suc  B )
3 rdgdmlim 7084 . . . . 5  |-  Lim  dom  rec ( ( x  e. 
_V  |->  C ) ,  A )
4 limsuc 6669 . . . . 5  |-  ( Lim 
dom  rec ( ( x  e.  _V  |->  C ) ,  A )  -> 
( B  e.  dom  rec ( ( x  e. 
_V  |->  C ) ,  A )  <->  suc  B  e. 
dom  rec ( ( x  e.  _V  |->  C ) ,  A ) ) )
53, 4ax-mp 5 . . . 4  |-  ( B  e.  dom  rec (
( x  e.  _V  |->  C ) ,  A
)  <->  suc  B  e.  dom  rec ( ( x  e. 
_V  |->  C ) ,  A ) )
6 rdgsucg 7090 . . . . . . 7  |-  ( B  e.  dom  rec (
( x  e.  _V  |->  C ) ,  A
)  ->  ( rec ( ( x  e. 
_V  |->  C ) ,  A ) `  suc  B )  =  ( ( x  e.  _V  |->  C ) `  ( rec ( ( x  e. 
_V  |->  C ) ,  A ) `  B
) ) )
71fveq1i 5867 . . . . . . . 8  |-  ( F `
 B )  =  ( rec ( ( x  e.  _V  |->  C ) ,  A ) `
 B )
87fveq2i 5869 . . . . . . 7  |-  ( ( x  e.  _V  |->  C ) `  ( F `
 B ) )  =  ( ( x  e.  _V  |->  C ) `
 ( rec (
( x  e.  _V  |->  C ) ,  A
) `  B )
)
96, 8syl6eqr 2526 . . . . . 6  |-  ( B  e.  dom  rec (
( x  e.  _V  |->  C ) ,  A
)  ->  ( rec ( ( x  e. 
_V  |->  C ) ,  A ) `  suc  B )  =  ( ( x  e.  _V  |->  C ) `  ( F `
 B ) ) )
10 nfmpt1 4536 . . . . . . . . . 10  |-  F/_ x
( x  e.  _V  |->  C )
11 rdgsucmptf.1 . . . . . . . . . 10  |-  F/_ x A
1210, 11nfrdg 7081 . . . . . . . . 9  |-  F/_ x rec ( ( x  e. 
_V  |->  C ) ,  A )
131, 12nfcxfr 2627 . . . . . . . 8  |-  F/_ x F
14 rdgsucmptf.2 . . . . . . . 8  |-  F/_ x B
1513, 14nffv 5873 . . . . . . 7  |-  F/_ x
( F `  B
)
16 rdgsucmptf.3 . . . . . . 7  |-  F/_ x D
17 rdgsucmptf.5 . . . . . . 7  |-  ( x  =  ( F `  B )  ->  C  =  D )
18 eqid 2467 . . . . . . 7  |-  ( x  e.  _V  |->  C )  =  ( x  e. 
_V  |->  C )
1915, 16, 17, 18fvmptnf 5968 . . . . . 6  |-  ( -.  D  e.  _V  ->  ( ( x  e.  _V  |->  C ) `  ( F `  B )
)  =  (/) )
209, 19sylan9eqr 2530 . . . . 5  |-  ( ( -.  D  e.  _V  /\  B  e.  dom  rec ( ( x  e. 
_V  |->  C ) ,  A ) )  -> 
( rec ( ( x  e.  _V  |->  C ) ,  A ) `
 suc  B )  =  (/) )
2120ex 434 . . . 4  |-  ( -.  D  e.  _V  ->  ( B  e.  dom  rec ( ( x  e. 
_V  |->  C ) ,  A )  ->  ( rec ( ( x  e. 
_V  |->  C ) ,  A ) `  suc  B )  =  (/) ) )
225, 21syl5bir 218 . . 3  |-  ( -.  D  e.  _V  ->  ( suc  B  e.  dom  rec ( ( x  e. 
_V  |->  C ) ,  A )  ->  ( rec ( ( x  e. 
_V  |->  C ) ,  A ) `  suc  B )  =  (/) ) )
23 ndmfv 5890 . . 3  |-  ( -. 
suc  B  e.  dom  rec ( ( x  e. 
_V  |->  C ) ,  A )  ->  ( rec ( ( x  e. 
_V  |->  C ) ,  A ) `  suc  B )  =  (/) )
2422, 23pm2.61d1 159 . 2  |-  ( -.  D  e.  _V  ->  ( rec ( ( x  e.  _V  |->  C ) ,  A ) `  suc  B )  =  (/) )
252, 24syl5eq 2520 1  |-  ( -.  D  e.  _V  ->  ( F `  suc  B
)  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    = wceq 1379    e. wcel 1767   F/_wnfc 2615   _Vcvv 3113   (/)c0 3785    |-> cmpt 4505   Lim wlim 4879   suc csuc 4880   dom cdm 4999   ` cfv 5588   reccrdg 7076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-recs 7043  df-rdg 7077
This theorem is referenced by: (None)
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