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Theorem rdgsucmptnf 7165
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 7164 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 5880 . 2  |-  ( F `
 suc  B )  =  ( rec (
( x  e.  _V  |->  C ) ,  A
) `  suc  B )
3 rdgdmlim 7153 . . . . 5  |-  Lim  dom  rec ( ( x  e. 
_V  |->  C ) ,  A )
4 limsuc 6695 . . . . 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 7159 . . . . . . 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 5880 . . . . . . . 8  |-  ( F `
 B )  =  ( rec ( ( x  e.  _V  |->  C ) ,  A ) `
 B )
87fveq2i 5882 . . . . . . 7  |-  ( ( x  e.  _V  |->  C ) `  ( F `
 B ) )  =  ( ( x  e.  _V  |->  C ) `
 ( rec (
( x  e.  _V  |->  C ) ,  A
) `  B )
)
96, 8syl6eqr 2523 . . . . . 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 4485 . . . . . . . . . 10  |-  F/_ x
( x  e.  _V  |->  C )
11 rdgsucmptf.1 . . . . . . . . . 10  |-  F/_ x A
1210, 11nfrdg 7150 . . . . . . . . 9  |-  F/_ x rec ( ( x  e. 
_V  |->  C ) ,  A )
131, 12nfcxfr 2610 . . . . . . . 8  |-  F/_ x F
14 rdgsucmptf.2 . . . . . . . 8  |-  F/_ x B
1513, 14nffv 5886 . . . . . . 7  |-  F/_ x
( F `  B
)
16 rdgsucmptf.3 . . . . . . 7  |-  F/_ x D
17 rdgsucmptf.5 . . . . . . 7  |-  ( x  =  ( F `  B )  ->  C  =  D )
18 eqid 2471 . . . . . . 7  |-  ( x  e.  _V  |->  C )  =  ( x  e. 
_V  |->  C )
1915, 16, 17, 18fvmptnf 5982 . . . . . 6  |-  ( -.  D  e.  _V  ->  ( ( x  e.  _V  |->  C ) `  ( F `  B )
)  =  (/) )
209, 19sylan9eqr 2527 . . . . 5  |-  ( ( -.  D  e.  _V  /\  B  e.  dom  rec ( ( x  e. 
_V  |->  C ) ,  A ) )  -> 
( rec ( ( x  e.  _V  |->  C ) ,  A ) `
 suc  B )  =  (/) )
2120ex 441 . . . 4  |-  ( -.  D  e.  _V  ->  ( B  e.  dom  rec ( ( x  e. 
_V  |->  C ) ,  A )  ->  ( rec ( ( x  e. 
_V  |->  C ) ,  A ) `  suc  B )  =  (/) ) )
225, 21syl5bir 226 . . 3  |-  ( -.  D  e.  _V  ->  ( suc  B  e.  dom  rec ( ( x  e. 
_V  |->  C ) ,  A )  ->  ( rec ( ( x  e. 
_V  |->  C ) ,  A ) `  suc  B )  =  (/) ) )
23 ndmfv 5903 . . 3  |-  ( -. 
suc  B  e.  dom  rec ( ( x  e. 
_V  |->  C ) ,  A )  ->  ( rec ( ( x  e. 
_V  |->  C ) ,  A ) `  suc  B )  =  (/) )
2422, 23pm2.61d1 164 . 2  |-  ( -.  D  e.  _V  ->  ( rec ( ( x  e.  _V  |->  C ) ,  A ) `  suc  B )  =  (/) )
252, 24syl5eq 2517 1  |-  ( -.  D  e.  _V  ->  ( F `  suc  B
)  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    = wceq 1452    e. wcel 1904   F/_wnfc 2599   _Vcvv 3031   (/)c0 3722    |-> cmpt 4454   dom cdm 4839   Lim wlim 5431   suc csuc 5432   ` cfv 5589   reccrdg 7145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-wrecs 7046  df-recs 7108  df-rdg 7146
This theorem is referenced by: (None)
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