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Theorem rdgsucmpt2 6995
Description: This version of rdgsucmpt 6996 avoids the not-free hypothesis of rdgsucmptf 6993 by using two substitutions instead of one. (Contributed by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
rdgsucmpt2.1  |-  F  =  rec ( ( x  e.  _V  |->  C ) ,  A )
rdgsucmpt2.2  |-  ( y  =  x  ->  E  =  C )
rdgsucmpt2.3  |-  ( y  =  ( F `  B )  ->  E  =  D )
Assertion
Ref Expression
rdgsucmpt2  |-  ( ( B  e.  On  /\  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 rdgsucmpt2
StepHypRef Expression
1 nfcv 2616 . 2  |-  F/_ y A
2 nfcv 2616 . 2  |-  F/_ y B
3 nfcv 2616 . 2  |-  F/_ y D
4 rdgsucmpt2.1 . . 3  |-  F  =  rec ( ( x  e.  _V  |->  C ) ,  A )
5 rdgsucmpt2.2 . . . . 5  |-  ( y  =  x  ->  E  =  C )
65cbvmptv 4490 . . . 4  |-  ( y  e.  _V  |->  E )  =  ( x  e. 
_V  |->  C )
7 rdgeq1 6976 . . . 4  |-  ( ( y  e.  _V  |->  E )  =  ( x  e.  _V  |->  C )  ->  rec ( ( y  e.  _V  |->  E ) ,  A )  =  rec ( ( x  e.  _V  |->  C ) ,  A ) )
86, 7ax-mp 5 . . 3  |-  rec (
( y  e.  _V  |->  E ) ,  A
)  =  rec (
( x  e.  _V  |->  C ) ,  A
)
94, 8eqtr4i 2486 . 2  |-  F  =  rec ( ( y  e.  _V  |->  E ) ,  A )
10 rdgsucmpt2.3 . 2  |-  ( y  =  ( F `  B )  ->  E  =  D )
111, 2, 3, 9, 10rdgsucmptf 6993 1  |-  ( ( B  e.  On  /\  D  e.  V )  ->  ( F `  suc  B )  =  D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3076    |-> cmpt 4457   Oncon0 4826   suc csuc 4828   ` cfv 5525   reccrdg 6974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-recs 6941  df-rdg 6975
This theorem is referenced by: (None)
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