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Theorem rdgsucmpt2 6655
Description: This version of rdgsucmpt 6656 avoids the not-free hypothesis of rdgsucmptf 6653 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 2548 . 2  |-  F/_ y A
2 nfcv 2548 . 2  |-  F/_ y B
3 nfcv 2548 . 2  |-  F/_ y D
4 rdgsucmpt2.1 . . 3  |-  F  =  rec ( ( x  e.  _V  |->  C ) ,  A )
5 rdgsucmpt2.2 . . . . 5  |-  ( y  =  x  ->  E  =  C )
65cbvmptv 4268 . . . 4  |-  ( y  e.  _V  |->  E )  =  ( x  e. 
_V  |->  C )
7 rdgeq1 6636 . . . 4  |-  ( ( y  e.  _V  |->  E )  =  ( x  e.  _V  |->  C )  ->  rec ( ( y  e.  _V  |->  E ) ,  A )  =  rec ( ( x  e.  _V  |->  C ) ,  A ) )
86, 7ax-mp 8 . . 3  |-  rec (
( y  e.  _V  |->  E ) ,  A
)  =  rec (
( x  e.  _V  |->  C ) ,  A
)
94, 8eqtr4i 2435 . 2  |-  F  =  rec ( ( y  e.  _V  |->  E ) ,  A )
10 rdgsucmpt2.3 . 2  |-  ( y  =  ( F `  B )  ->  E  =  D )
111, 2, 3, 9, 10rdgsucmptf 6653 1  |-  ( ( B  e.  On  /\  D  e.  V )  ->  ( F `  suc  B )  =  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2924    e. cmpt 4234   Oncon0 4549   suc csuc 4551   ` cfv 5421   reccrdg 6634
This theorem is referenced by:  abianfplem  6682
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-recs 6600  df-rdg 6635
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