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Theorem rdgsucmptf 6678
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 6674 . . 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 5721 . . 3  |-  ( F `
 suc  B )  =  ( rec (
( x  e.  _V  |->  C ) ,  A
) `  suc  B )
42fveq1i 5721 . . . 4  |-  ( F `
 B )  =  ( rec ( ( x  e.  _V  |->  C ) ,  A ) `
 B )
54fveq2i 5723 . . 3  |-  ( ( x  e.  _V  |->  C ) `  ( F `
 B ) )  =  ( ( x  e.  _V  |->  C ) `
 ( rec (
( x  e.  _V  |->  C ) ,  A
) `  B )
)
61, 3, 53eqtr4g 2492 . 2  |-  ( B  e.  On  ->  ( F `  suc  B )  =  ( ( x  e.  _V  |->  C ) `
 ( F `  B ) ) )
7 fvex 5734 . . 3  |-  ( F `
 B )  e. 
_V
8 nfmpt1 4290 . . . . . . 7  |-  F/_ x
( x  e.  _V  |->  C )
9 rdgsucmptf.1 . . . . . . 7  |-  F/_ x A
108, 9nfrdg 6664 . . . . . 6  |-  F/_ x rec ( ( x  e. 
_V  |->  C ) ,  A )
112, 10nfcxfr 2568 . . . . 5  |-  F/_ x F
12 rdgsucmptf.2 . . . . 5  |-  F/_ x B
1311, 12nffv 5727 . . . 4  |-  F/_ x
( F `  B
)
14 rdgsucmptf.3 . . . 4  |-  F/_ x D
15 rdgsucmptf.5 . . . 4  |-  ( x  =  ( F `  B )  ->  C  =  D )
16 eqid 2435 . . . 4  |-  ( x  e.  _V  |->  C )  =  ( x  e. 
_V  |->  C )
1713, 14, 15, 16fvmptf 5813 . . 3  |-  ( ( ( F `  B
)  e.  _V  /\  D  e.  V )  ->  ( ( x  e. 
_V  |->  C ) `  ( F `  B ) )  =  D )
187, 17mpan 652 . 2  |-  ( D  e.  V  ->  (
( x  e.  _V  |->  C ) `  ( F `  B )
)  =  D )
196, 18sylan9eq 2487 1  |-  ( ( B  e.  On  /\  D  e.  V )  ->  ( F `  suc  B )  =  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   F/_wnfc 2558   _Vcvv 2948    e. cmpt 4258   Oncon0 4573   suc csuc 4575   ` cfv 5446   reccrdg 6659
This theorem is referenced by:  rdgsucmpt2  6680  rdgsucmpt  6681
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-recs 6625  df-rdg 6660
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