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Theorem rdgsucmptnf 6687
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 6686 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 5729 . 2  |-  ( F `
 suc  B )  =  ( rec (
( x  e.  _V  |->  C ) ,  A
) `  suc  B )
3 rdgdmlim 6675 . . . . 5  |-  Lim  dom  rec ( ( x  e. 
_V  |->  C ) ,  A )
4 limsuc 4829 . . . . 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 8 . . . 4  |-  ( B  e.  dom  rec (
( x  e.  _V  |->  C ) ,  A
)  <->  suc  B  e.  dom  rec ( ( x  e. 
_V  |->  C ) ,  A ) )
6 rdgsucg 6681 . . . . . . 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 5729 . . . . . . . 8  |-  ( F `
 B )  =  ( rec ( ( x  e.  _V  |->  C ) ,  A ) `
 B )
87fveq2i 5731 . . . . . . 7  |-  ( ( x  e.  _V  |->  C ) `  ( F `
 B ) )  =  ( ( x  e.  _V  |->  C ) `
 ( rec (
( x  e.  _V  |->  C ) ,  A
) `  B )
)
96, 8syl6eqr 2486 . . . . . 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 4298 . . . . . . . . . 10  |-  F/_ x
( x  e.  _V  |->  C )
11 rdgsucmptf.1 . . . . . . . . . 10  |-  F/_ x A
1210, 11nfrdg 6672 . . . . . . . . 9  |-  F/_ x rec ( ( x  e. 
_V  |->  C ) ,  A )
131, 12nfcxfr 2569 . . . . . . . 8  |-  F/_ x F
14 rdgsucmptf.2 . . . . . . . 8  |-  F/_ x B
1513, 14nffv 5735 . . . . . . 7  |-  F/_ x
( F `  B
)
16 rdgsucmptf.3 . . . . . . 7  |-  F/_ x D
17 rdgsucmptf.5 . . . . . . 7  |-  ( x  =  ( F `  B )  ->  C  =  D )
18 eqid 2436 . . . . . . 7  |-  ( x  e.  _V  |->  C )  =  ( x  e. 
_V  |->  C )
1915, 16, 17, 18fvmptnf 5822 . . . . . 6  |-  ( -.  D  e.  _V  ->  ( ( x  e.  _V  |->  C ) `  ( F `  B )
)  =  (/) )
209, 19sylan9eqr 2490 . . . . 5  |-  ( ( -.  D  e.  _V  /\  B  e.  dom  rec ( ( x  e. 
_V  |->  C ) ,  A ) )  -> 
( rec ( ( x  e.  _V  |->  C ) ,  A ) `
 suc  B )  =  (/) )
2120ex 424 . . . 4  |-  ( -.  D  e.  _V  ->  ( B  e.  dom  rec ( ( x  e. 
_V  |->  C ) ,  A )  ->  ( rec ( ( x  e. 
_V  |->  C ) ,  A ) `  suc  B )  =  (/) ) )
225, 21syl5bir 210 . . 3  |-  ( -.  D  e.  _V  ->  ( suc  B  e.  dom  rec ( ( x  e. 
_V  |->  C ) ,  A )  ->  ( rec ( ( x  e. 
_V  |->  C ) ,  A ) `  suc  B )  =  (/) ) )
23 ndmfv 5755 . . 3  |-  ( -. 
suc  B  e.  dom  rec ( ( x  e. 
_V  |->  C ) ,  A )  ->  ( rec ( ( x  e. 
_V  |->  C ) ,  A ) `  suc  B )  =  (/) )
2422, 23pm2.61d1 153 . 2  |-  ( -.  D  e.  _V  ->  ( rec ( ( x  e.  _V  |->  C ) ,  A ) `  suc  B )  =  (/) )
252, 24syl5eq 2480 1  |-  ( -.  D  e.  _V  ->  ( F `  suc  B
)  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725   F/_wnfc 2559   _Vcvv 2956   (/)c0 3628    e. cmpt 4266   Lim wlim 4582   suc csuc 4583   dom cdm 4878   ` cfv 5454   reccrdg 6667
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-recs 6633  df-rdg 6668
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