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Theorem rdgsucmptf 6457
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 6453 . . 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 5542 . . 3  |-  ( F `
 suc  B )  =  ( rec (
( x  e.  _V  |->  C ) ,  A
) `  suc  B )
42fveq1i 5542 . . . 4  |-  ( F `
 B )  =  ( rec ( ( x  e.  _V  |->  C ) ,  A ) `
 B )
54fveq2i 5544 . . 3  |-  ( ( x  e.  _V  |->  C ) `  ( F `
 B ) )  =  ( ( x  e.  _V  |->  C ) `
 ( rec (
( x  e.  _V  |->  C ) ,  A
) `  B )
)
61, 3, 53eqtr4g 2353 . 2  |-  ( B  e.  On  ->  ( F `  suc  B )  =  ( ( x  e.  _V  |->  C ) `
 ( F `  B ) ) )
7 fvex 5555 . . 3  |-  ( F `
 B )  e. 
_V
8 nfmpt1 4125 . . . . . . 7  |-  F/_ x
( x  e.  _V  |->  C )
9 rdgsucmptf.1 . . . . . . 7  |-  F/_ x A
108, 9nfrdg 6443 . . . . . 6  |-  F/_ x rec ( ( x  e. 
_V  |->  C ) ,  A )
112, 10nfcxfr 2429 . . . . 5  |-  F/_ x F
12 rdgsucmptf.2 . . . . 5  |-  F/_ x B
1311, 12nffv 5548 . . . 4  |-  F/_ x
( F `  B
)
14 rdgsucmptf.3 . . . 4  |-  F/_ x D
15 rdgsucmptf.5 . . . 4  |-  ( x  =  ( F `  B )  ->  C  =  D )
16 eqid 2296 . . . 4  |-  ( x  e.  _V  |->  C )  =  ( x  e. 
_V  |->  C )
1713, 14, 15, 16fvmptf 5632 . . 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 2348 1  |-  ( ( B  e.  On  /\  D  e.  V )  ->  ( F `  suc  B )  =  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   F/_wnfc 2419   _Vcvv 2801    e. cmpt 4093   Oncon0 4408   suc csuc 4410   ` cfv 5271   reccrdg 6438
This theorem is referenced by:  rdgsucmpt2  6459  rdgsucmpt  6460
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-recs 6404  df-rdg 6439
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