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Theorem rdgsucmptnf 7095
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 7094 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 5819 . 2  |-  ( F `
 suc  B )  =  ( rec (
( x  e.  _V  |->  C ) ,  A
) `  suc  B )
3 rdgdmlim 7083 . . . . 5  |-  Lim  dom  rec ( ( x  e. 
_V  |->  C ) ,  A )
4 limsuc 6627 . . . . 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 7089 . . . . . . 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 5819 . . . . . . . 8  |-  ( F `
 B )  =  ( rec ( ( x  e.  _V  |->  C ) ,  A ) `
 B )
87fveq2i 5821 . . . . . . 7  |-  ( ( x  e.  _V  |->  C ) `  ( F `
 B ) )  =  ( ( x  e.  _V  |->  C ) `
 ( rec (
( x  e.  _V  |->  C ) ,  A
) `  B )
)
96, 8syl6eqr 2474 . . . . . 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 4449 . . . . . . . . . 10  |-  F/_ x
( x  e.  _V  |->  C )
11 rdgsucmptf.1 . . . . . . . . . 10  |-  F/_ x A
1210, 11nfrdg 7080 . . . . . . . . 9  |-  F/_ x rec ( ( x  e. 
_V  |->  C ) ,  A )
131, 12nfcxfr 2561 . . . . . . . 8  |-  F/_ x F
14 rdgsucmptf.2 . . . . . . . 8  |-  F/_ x B
1513, 14nffv 5825 . . . . . . 7  |-  F/_ x
( F `  B
)
16 rdgsucmptf.3 . . . . . . 7  |-  F/_ x D
17 rdgsucmptf.5 . . . . . . 7  |-  ( x  =  ( F `  B )  ->  C  =  D )
18 eqid 2422 . . . . . . 7  |-  ( x  e.  _V  |->  C )  =  ( x  e. 
_V  |->  C )
1915, 16, 17, 18fvmptnf 5920 . . . . . 6  |-  ( -.  D  e.  _V  ->  ( ( x  e.  _V  |->  C ) `  ( F `  B )
)  =  (/) )
209, 19sylan9eqr 2478 . . . . 5  |-  ( ( -.  D  e.  _V  /\  B  e.  dom  rec ( ( x  e. 
_V  |->  C ) ,  A ) )  -> 
( rec ( ( x  e.  _V  |->  C ) ,  A ) `
 suc  B )  =  (/) )
2120ex 435 . . . 4  |-  ( -.  D  e.  _V  ->  ( B  e.  dom  rec ( ( x  e. 
_V  |->  C ) ,  A )  ->  ( rec ( ( x  e. 
_V  |->  C ) ,  A ) `  suc  B )  =  (/) ) )
225, 21syl5bir 221 . . 3  |-  ( -.  D  e.  _V  ->  ( suc  B  e.  dom  rec ( ( x  e. 
_V  |->  C ) ,  A )  ->  ( rec ( ( x  e. 
_V  |->  C ) ,  A ) `  suc  B )  =  (/) ) )
23 ndmfv 5842 . . 3  |-  ( -. 
suc  B  e.  dom  rec ( ( x  e. 
_V  |->  C ) ,  A )  ->  ( rec ( ( x  e. 
_V  |->  C ) ,  A ) `  suc  B )  =  (/) )
2422, 23pm2.61d1 162 . 2  |-  ( -.  D  e.  _V  ->  ( rec ( ( x  e.  _V  |->  C ) ,  A ) `  suc  B )  =  (/) )
252, 24syl5eq 2468 1  |-  ( -.  D  e.  _V  ->  ( F `  suc  B
)  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    = wceq 1437    e. wcel 1872   F/_wnfc 2550   _Vcvv 3016   (/)c0 3697    |-> cmpt 4418   dom cdm 4789   Lim wlim 5379   suc csuc 5380   ` cfv 5537   reccrdg 7075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2402  ax-sep 4482  ax-nul 4491  ax-pow 4538  ax-pr 4596  ax-un 6534
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2552  df-ne 2595  df-ral 2713  df-rex 2714  df-reu 2715  df-rab 2717  df-v 3018  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-pss 3388  df-nul 3698  df-if 3848  df-pw 3919  df-sn 3935  df-pr 3937  df-tp 3939  df-op 3941  df-uni 4156  df-iun 4237  df-br 4360  df-opab 4419  df-mpt 4420  df-tr 4455  df-eprel 4700  df-id 4704  df-po 4710  df-so 4711  df-fr 4748  df-we 4750  df-xp 4795  df-rel 4796  df-cnv 4797  df-co 4798  df-dm 4799  df-rn 4800  df-res 4801  df-ima 4802  df-pred 5335  df-ord 5381  df-on 5382  df-lim 5383  df-suc 5384  df-iota 5501  df-fun 5539  df-fn 5540  df-f 5541  df-f1 5542  df-fo 5543  df-f1o 5544  df-fv 5545  df-wrecs 6976  df-recs 7038  df-rdg 7076
This theorem is referenced by: (None)
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