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Theorem frsucmpt2 6468
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 2432 . 2  |-  F/_ y A
2 nfcv 2432 . 2  |-  F/_ y B
3 nfcv 2432 . 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 4127 . . . . 5  |-  ( y  e.  _V  |->  E )  =  ( x  e. 
_V  |->  C )
7 rdgeq1 6440 . . . . 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 4967 . . 3  |-  ( rec ( ( y  e. 
_V  |->  E ) ,  A )  |`  om )  =  ( rec (
( x  e.  _V  |->  C ) ,  A
)  |`  om )
104, 9eqtr4i 2319 . 2  |-  F  =  ( rec ( ( y  e.  _V  |->  E ) ,  A )  |`  om )
11 frsucmpt2.3 . 2  |-  ( y  =  ( F `  B )  ->  E  =  D )
121, 2, 3, 10, 11frsucmpt 6466 1  |-  ( ( B  e.  om  /\  D  e.  V )  ->  ( F `  suc  B )  =  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    e. cmpt 4093   suc csuc 4410   omcom 4672    |` cres 4707   ` cfv 5271   reccrdg 6438
This theorem is referenced by:  unblem2  7126  unblem3  7127  inf0  7338  trcl  7426  hsmexlem8  8066  wunex2  8376  wuncval2  8385  peano5nni  9765  peano2nn  9774  om2uzsuci  11027  expm  25467  neibastop2lem  26412
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-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-om 4673  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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