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Theorem rdgsucmptf 7152
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 7148 . . 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 5880 . . 3  |-  ( F `
 suc  B )  =  ( rec (
( x  e.  _V  |->  C ) ,  A
) `  suc  B )
42fveq1i 5880 . . . 4  |-  ( F `
 B )  =  ( rec ( ( x  e.  _V  |->  C ) ,  A ) `
 B )
54fveq2i 5882 . . 3  |-  ( ( x  e.  _V  |->  C ) `  ( F `
 B ) )  =  ( ( x  e.  _V  |->  C ) `
 ( rec (
( x  e.  _V  |->  C ) ,  A
) `  B )
)
61, 3, 53eqtr4g 2489 . 2  |-  ( B  e.  On  ->  ( F `  suc  B )  =  ( ( x  e.  _V  |->  C ) `
 ( F `  B ) ) )
7 fvex 5889 . . 3  |-  ( F `
 B )  e. 
_V
8 nfmpt1 4511 . . . . . . 7  |-  F/_ x
( x  e.  _V  |->  C )
9 rdgsucmptf.1 . . . . . . 7  |-  F/_ x A
108, 9nfrdg 7138 . . . . . 6  |-  F/_ x rec ( ( x  e. 
_V  |->  C ) ,  A )
112, 10nfcxfr 2583 . . . . 5  |-  F/_ x F
12 rdgsucmptf.2 . . . . 5  |-  F/_ x B
1311, 12nffv 5886 . . . 4  |-  F/_ x
( F `  B
)
14 rdgsucmptf.3 . . . 4  |-  F/_ x D
15 rdgsucmptf.5 . . . 4  |-  ( x  =  ( F `  B )  ->  C  =  D )
16 eqid 2423 . . . 4  |-  ( x  e.  _V  |->  C )  =  ( x  e. 
_V  |->  C )
1713, 14, 15, 16fvmptf 5980 . . 3  |-  ( ( ( F `  B
)  e.  _V  /\  D  e.  V )  ->  ( ( x  e. 
_V  |->  C ) `  ( F `  B ) )  =  D )
187, 17mpan 675 . 2  |-  ( D  e.  V  ->  (
( x  e.  _V  |->  C ) `  ( F `  B )
)  =  D )
196, 18sylan9eq 2484 1  |-  ( ( B  e.  On  /\  D  e.  V )  ->  ( F `  suc  B )  =  D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1438    e. wcel 1869   F/_wnfc 2571   _Vcvv 3082    |-> cmpt 4480   Oncon0 5440   suc csuc 5442   ` cfv 5599   reccrdg 7133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-wrecs 7034  df-recs 7096  df-rdg 7134
This theorem is referenced by:  rdgsucmpt2  7154  rdgsucmpt  7155
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