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Theorem rdgsucmptf 6995
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 6991 . . 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 5801 . . 3  |-  ( F `
 suc  B )  =  ( rec (
( x  e.  _V  |->  C ) ,  A
) `  suc  B )
42fveq1i 5801 . . . 4  |-  ( F `
 B )  =  ( rec ( ( x  e.  _V  |->  C ) ,  A ) `
 B )
54fveq2i 5803 . . 3  |-  ( ( x  e.  _V  |->  C ) `  ( F `
 B ) )  =  ( ( x  e.  _V  |->  C ) `
 ( rec (
( x  e.  _V  |->  C ) ,  A
) `  B )
)
61, 3, 53eqtr4g 2520 . 2  |-  ( B  e.  On  ->  ( F `  suc  B )  =  ( ( x  e.  _V  |->  C ) `
 ( F `  B ) ) )
7 fvex 5810 . . 3  |-  ( F `
 B )  e. 
_V
8 nfmpt1 4490 . . . . . . 7  |-  F/_ x
( x  e.  _V  |->  C )
9 rdgsucmptf.1 . . . . . . 7  |-  F/_ x A
108, 9nfrdg 6981 . . . . . 6  |-  F/_ x rec ( ( x  e. 
_V  |->  C ) ,  A )
112, 10nfcxfr 2614 . . . . 5  |-  F/_ x F
12 rdgsucmptf.2 . . . . 5  |-  F/_ x B
1311, 12nffv 5807 . . . 4  |-  F/_ x
( F `  B
)
14 rdgsucmptf.3 . . . 4  |-  F/_ x D
15 rdgsucmptf.5 . . . 4  |-  ( x  =  ( F `  B )  ->  C  =  D )
16 eqid 2454 . . . 4  |-  ( x  e.  _V  |->  C )  =  ( x  e. 
_V  |->  C )
1713, 14, 15, 16fvmptf 5900 . . 3  |-  ( ( ( F `  B
)  e.  _V  /\  D  e.  V )  ->  ( ( x  e. 
_V  |->  C ) `  ( F `  B ) )  =  D )
187, 17mpan 670 . 2  |-  ( D  e.  V  ->  (
( x  e.  _V  |->  C ) `  ( F `  B )
)  =  D )
196, 18sylan9eq 2515 1  |-  ( ( B  e.  On  /\  D  e.  V )  ->  ( F `  suc  B )  =  D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   F/_wnfc 2602   _Vcvv 3078    |-> cmpt 4459   Oncon0 4828   suc csuc 4830   ` cfv 5527   reccrdg 6976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-recs 6943  df-rdg 6977
This theorem is referenced by:  rdgsucmpt2  6997  rdgsucmpt  6998
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