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Theorem rdgsucmptnf 7113
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 7112 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 5873 . 2  |-  ( F `
 suc  B )  =  ( rec (
( x  e.  _V  |->  C ) ,  A
) `  suc  B )
3 rdgdmlim 7101 . . . . 5  |-  Lim  dom  rec ( ( x  e. 
_V  |->  C ) ,  A )
4 limsuc 6683 . . . . 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 7107 . . . . . . 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 5873 . . . . . . . 8  |-  ( F `
 B )  =  ( rec ( ( x  e.  _V  |->  C ) ,  A ) `
 B )
87fveq2i 5875 . . . . . . 7  |-  ( ( x  e.  _V  |->  C ) `  ( F `
 B ) )  =  ( ( x  e.  _V  |->  C ) `
 ( rec (
( x  e.  _V  |->  C ) ,  A
) `  B )
)
96, 8syl6eqr 2516 . . . . . 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 4546 . . . . . . . . . 10  |-  F/_ x
( x  e.  _V  |->  C )
11 rdgsucmptf.1 . . . . . . . . . 10  |-  F/_ x A
1210, 11nfrdg 7098 . . . . . . . . 9  |-  F/_ x rec ( ( x  e. 
_V  |->  C ) ,  A )
131, 12nfcxfr 2617 . . . . . . . 8  |-  F/_ x F
14 rdgsucmptf.2 . . . . . . . 8  |-  F/_ x B
1513, 14nffv 5879 . . . . . . 7  |-  F/_ x
( F `  B
)
16 rdgsucmptf.3 . . . . . . 7  |-  F/_ x D
17 rdgsucmptf.5 . . . . . . 7  |-  ( x  =  ( F `  B )  ->  C  =  D )
18 eqid 2457 . . . . . . 7  |-  ( x  e.  _V  |->  C )  =  ( x  e. 
_V  |->  C )
1915, 16, 17, 18fvmptnf 5974 . . . . . 6  |-  ( -.  D  e.  _V  ->  ( ( x  e.  _V  |->  C ) `  ( F `  B )
)  =  (/) )
209, 19sylan9eqr 2520 . . . . 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 5896 . . 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 2510 1  |-  ( -.  D  e.  _V  ->  ( F `  suc  B
)  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    = wceq 1395    e. wcel 1819   F/_wnfc 2605   _Vcvv 3109   (/)c0 3793    |-> cmpt 4515   Lim wlim 4888   suc csuc 4889   dom cdm 5008   ` cfv 5594   reccrdg 7093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-recs 7060  df-rdg 7094
This theorem is referenced by: (None)
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