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Theorem frsucmpt2 6907
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 2589 . 2  |-  F/_ y A
2 nfcv 2589 . 2  |-  F/_ y B
3 nfcv 2589 . 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 4395 . . . . 5  |-  ( y  e.  _V  |->  E )  =  ( x  e. 
_V  |->  C )
7 rdgeq1 6879 . . . . 5  |-  ( ( y  e.  _V  |->  E )  =  ( x  e.  _V  |->  C )  ->  rec ( ( y  e.  _V  |->  E ) ,  A )  =  rec ( ( x  e.  _V  |->  C ) ,  A ) )
86, 7ax-mp 5 . . . 4  |-  rec (
( y  e.  _V  |->  E ) ,  A
)  =  rec (
( x  e.  _V  |->  C ) ,  A
)
98reseq1i 5118 . . 3  |-  ( rec ( ( y  e. 
_V  |->  E ) ,  A )  |`  om )  =  ( rec (
( x  e.  _V  |->  C ) ,  A
)  |`  om )
104, 9eqtr4i 2466 . 2  |-  F  =  ( rec ( ( y  e.  _V  |->  E ) ,  A )  |`  om )
11 frsucmpt2.3 . 2  |-  ( y  =  ( F `  B )  ->  E  =  D )
121, 2, 3, 10, 11frsucmpt 6905 1  |-  ( ( B  e.  om  /\  D  e.  V )  ->  ( F `  suc  B )  =  D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2984    e. cmpt 4362   suc csuc 4733    |` cres 4854   ` cfv 5430   omcom 6488   reccrdg 6877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-om 6489  df-recs 6844  df-rdg 6878
This theorem is referenced by:  unblem2  7577  unblem3  7578  inf0  7839  trcl  7960  hsmexlem8  8605  wunex2  8917  wuncval2  8926  peano5nni  10337  peano2nn  10346  om2uzsuci  11783  neibastop2lem  28593
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