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Theorem frsucmpt2 6633
Description: The successor value resulting from finite recursive definition generation (special case where the generation function is expressed in maps-to notation), using double-substitution instead of a bound variable condition. (Contributed by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
frsucmpt2.1  |-  F  =  ( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om )
frsucmpt2.2  |-  ( y  =  x  ->  E  =  C )
frsucmpt2.3  |-  ( y  =  ( F `  B )  ->  E  =  D )
Assertion
Ref Expression
frsucmpt2  |-  ( ( B  e.  om  /\  D  e.  V )  ->  ( F `  suc  B )  =  D )
Distinct variable groups:    y, A    y, B    y, C    y, D    x, E
Allowed substitution hints:    A( x)    B( x)    C( x)    D( x)    E( y)    F( x, y)    V( x, y)

Proof of Theorem frsucmpt2
StepHypRef Expression
1 nfcv 2523 . 2  |-  F/_ y A
2 nfcv 2523 . 2  |-  F/_ y B
3 nfcv 2523 . 2  |-  F/_ y D
4 frsucmpt2.1 . . 3  |-  F  =  ( rec ( ( x  e.  _V  |->  C ) ,  A )  |`  om )
5 frsucmpt2.2 . . . . . 6  |-  ( y  =  x  ->  E  =  C )
65cbvmptv 4241 . . . . 5  |-  ( y  e.  _V  |->  E )  =  ( x  e. 
_V  |->  C )
7 rdgeq1 6605 . . . . 5  |-  ( ( y  e.  _V  |->  E )  =  ( x  e.  _V  |->  C )  ->  rec ( ( y  e.  _V  |->  E ) ,  A )  =  rec ( ( x  e.  _V  |->  C ) ,  A ) )
86, 7ax-mp 8 . . . 4  |-  rec (
( y  e.  _V  |->  E ) ,  A
)  =  rec (
( x  e.  _V  |->  C ) ,  A
)
98reseq1i 5082 . . 3  |-  ( rec ( ( y  e. 
_V  |->  E ) ,  A )  |`  om )  =  ( rec (
( x  e.  _V  |->  C ) ,  A
)  |`  om )
104, 9eqtr4i 2410 . 2  |-  F  =  ( rec ( ( y  e.  _V  |->  E ) ,  A )  |`  om )
11 frsucmpt2.3 . 2  |-  ( y  =  ( F `  B )  ->  E  =  D )
121, 2, 3, 10, 11frsucmpt 6631 1  |-  ( ( B  e.  om  /\  D  e.  V )  ->  ( F `  suc  B )  =  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2899    e. cmpt 4207   suc csuc 4524   omcom 4785    |` cres 4820   ` cfv 5394   reccrdg 6603
This theorem is referenced by:  unblem2  7296  unblem3  7297  inf0  7509  trcl  7597  hsmexlem8  8237  wunex2  8546  wuncval2  8555  peano5nni  9935  peano2nn  9944  om2uzsuci  11215  neibastop2lem  26080
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-recs 6569  df-rdg 6604
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