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Theorem frsucmpt2 6689
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 2571 . 2  |-  F/_ y A
2 nfcv 2571 . 2  |-  F/_ y B
3 nfcv 2571 . 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 4292 . . . . 5  |-  ( y  e.  _V  |->  E )  =  ( x  e. 
_V  |->  C )
7 rdgeq1 6661 . . . . 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 5134 . . 3  |-  ( rec ( ( y  e. 
_V  |->  E ) ,  A )  |`  om )  =  ( rec (
( x  e.  _V  |->  C ) ,  A
)  |`  om )
104, 9eqtr4i 2458 . 2  |-  F  =  ( rec ( ( y  e.  _V  |->  E ) ,  A )  |`  om )
11 frsucmpt2.3 . 2  |-  ( y  =  ( F `  B )  ->  E  =  D )
121, 2, 3, 10, 11frsucmpt 6687 1  |-  ( ( B  e.  om  /\  D  e.  V )  ->  ( F `  suc  B )  =  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948    e. cmpt 4258   suc csuc 4575   omcom 4837    |` cres 4872   ` cfv 5446   reccrdg 6659
This theorem is referenced by:  unblem2  7352  unblem3  7353  inf0  7568  trcl  7656  hsmexlem8  8296  wunex2  8605  wuncval2  8614  peano5nni  9995  peano2nn  10004  om2uzsuci  11280  neibastop2lem  26370
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-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-om 4838  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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