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Theorem rdgsucmptf 6441
Description: The value of the recursive definition generator at a successor (special case where the characteristic function uses the map operation). (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
rdgsucmptf  |-  ( ( B  e.  On  /\  D  e.  V )  ->  ( F `  suc  B )  =  D )

Proof of Theorem rdgsucmptf
StepHypRef Expression
1 rdgsuc 6437 . . 3  |-  ( B  e.  On  ->  ( rec ( ( x  e. 
_V  |->  C ) ,  A ) `  suc  B )  =  ( ( x  e.  _V  |->  C ) `  ( rec ( ( x  e. 
_V  |->  C ) ,  A ) `  B
) ) )
2 rdgsucmptf.4 . . . 4  |-  F  =  rec ( ( x  e.  _V  |->  C ) ,  A )
32fveq1i 5526 . . 3  |-  ( F `
 suc  B )  =  ( rec (
( x  e.  _V  |->  C ) ,  A
) `  suc  B )
42fveq1i 5526 . . . 4  |-  ( F `
 B )  =  ( rec ( ( x  e.  _V  |->  C ) ,  A ) `
 B )
54fveq2i 5528 . . 3  |-  ( ( x  e.  _V  |->  C ) `  ( F `
 B ) )  =  ( ( x  e.  _V  |->  C ) `
 ( rec (
( x  e.  _V  |->  C ) ,  A
) `  B )
)
61, 3, 53eqtr4g 2340 . 2  |-  ( B  e.  On  ->  ( F `  suc  B )  =  ( ( x  e.  _V  |->  C ) `
 ( F `  B ) ) )
7 fvex 5539 . . 3  |-  ( F `
 B )  e. 
_V
8 nfmpt1 4109 . . . . . . 7  |-  F/_ x
( x  e.  _V  |->  C )
9 rdgsucmptf.1 . . . . . . 7  |-  F/_ x A
108, 9nfrdg 6427 . . . . . 6  |-  F/_ x rec ( ( x  e. 
_V  |->  C ) ,  A )
112, 10nfcxfr 2416 . . . . 5  |-  F/_ x F
12 rdgsucmptf.2 . . . . 5  |-  F/_ x B
1311, 12nffv 5532 . . . 4  |-  F/_ x
( F `  B
)
14 rdgsucmptf.3 . . . 4  |-  F/_ x D
15 rdgsucmptf.5 . . . 4  |-  ( x  =  ( F `  B )  ->  C  =  D )
16 eqid 2283 . . . 4  |-  ( x  e.  _V  |->  C )  =  ( x  e. 
_V  |->  C )
1713, 14, 15, 16fvmptf 5616 . . 3  |-  ( ( ( F `  B
)  e.  _V  /\  D  e.  V )  ->  ( ( x  e. 
_V  |->  C ) `  ( F `  B ) )  =  D )
187, 17mpan 651 . 2  |-  ( D  e.  V  ->  (
( x  e.  _V  |->  C ) `  ( F `  B )
)  =  D )
196, 18sylan9eq 2335 1  |-  ( ( B  e.  On  /\  D  e.  V )  ->  ( F `  suc  B )  =  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   F/_wnfc 2406   _Vcvv 2788    e. cmpt 4077   Oncon0 4392   suc csuc 4394   ` cfv 5255   reccrdg 6422
This theorem is referenced by:  rdgsucmpt2  6443  rdgsucmpt  6444
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388  df-rdg 6423
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