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Theorem frsucmpt2 7106
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 2629 . 2  |-  F/_ y A
2 nfcv 2629 . 2  |-  F/_ y B
3 nfcv 2629 . 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 4538 . . . . 5  |-  ( y  e.  _V  |->  E )  =  ( x  e. 
_V  |->  C )
7 rdgeq1 7078 . . . . 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 5269 . . 3  |-  ( rec ( ( y  e. 
_V  |->  E ) ,  A )  |`  om )  =  ( rec (
( x  e.  _V  |->  C ) ,  A
)  |`  om )
104, 9eqtr4i 2499 . 2  |-  F  =  ( rec ( ( y  e.  _V  |->  E ) ,  A )  |`  om )
11 frsucmpt2.3 . 2  |-  ( y  =  ( F `  B )  ->  E  =  D )
121, 2, 3, 10, 11frsucmpt 7104 1  |-  ( ( B  e.  om  /\  D  e.  V )  ->  ( F `  suc  B )  =  D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113    |-> cmpt 4505   suc csuc 4880    |` cres 5001   ` cfv 5588   omcom 6685   reccrdg 7076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-om 6686  df-recs 7043  df-rdg 7077
This theorem is referenced by:  unblem2  7774  unblem3  7775  inf0  8039  trcl  8160  hsmexlem8  8805  wunex2  9117  wuncval2  9126  peano5nni  10540  peano2nn  10549  om2uzsuci  12028  neibastop2lem  30008
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