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Theorem ovmpt2dv2 5713
Description: Alternate deduction version of ovmpt2 5715, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
Hypotheses
Ref Expression
ovmpt2dv2.1  |-  ( ph  ->  A  e.  C )
ovmpt2dv2.2  |-  ( (
ph  /\  x  =  A )  ->  B  e.  D )
ovmpt2dv2.3  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  R  e.  V )
ovmpt2dv2.4  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  R  =  S )
Assertion
Ref Expression
ovmpt2dv2  |-  ( ph  ->  ( F  =  ( x  e.  C , 
y  e.  D  |->  R )  ->  ( A F B )  =  S ) )
Distinct variable groups:    x, y, A   
x, B, y    ph, x, y    x, S, y
Allowed substitution hints:    C( x, y)    D( x, y)    R( x, y)    F( x, y)    V( x, y)

Proof of Theorem ovmpt2dv2
StepHypRef Expression
1 eqidd 2084 . . 3  |-  ( ph  ->  ( x  e.  C ,  y  e.  D  |->  R )  =  ( x  e.  C , 
y  e.  D  |->  R ) )
2 ovmpt2dv2.1 . . . 4  |-  ( ph  ->  A  e.  C )
3 ovmpt2dv2.2 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  B  e.  D )
4 ovmpt2dv2.3 . . . 4  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  R  e.  V )
5 ovmpt2dv2.4 . . . . . 6  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  R  =  S )
65eqeq2d 2094 . . . . 5  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( ( A ( x  e.  C , 
y  e.  D  |->  R ) B )  =  R  <->  ( A ( x  e.  C , 
y  e.  D  |->  R ) B )  =  S ) )
76biimpd 142 . . . 4  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( ( A ( x  e.  C , 
y  e.  D  |->  R ) B )  =  R  ->  ( A
( x  e.  C ,  y  e.  D  |->  R ) B )  =  S ) )
8 nfmpt21 5650 . . . 4  |-  F/_ x
( x  e.  C ,  y  e.  D  |->  R )
9 nfcv 2223 . . . . . 6  |-  F/_ x A
10 nfcv 2223 . . . . . 6  |-  F/_ x B
119, 8, 10nfov 5614 . . . . 5  |-  F/_ x
( A ( x  e.  C ,  y  e.  D  |->  R ) B )
1211nfeq1 2232 . . . 4  |-  F/ x
( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S
13 nfmpt22 5651 . . . 4  |-  F/_ y
( x  e.  C ,  y  e.  D  |->  R )
14 nfcv 2223 . . . . . 6  |-  F/_ y A
15 nfcv 2223 . . . . . 6  |-  F/_ y B
1614, 13, 15nfov 5614 . . . . 5  |-  F/_ y
( A ( x  e.  C ,  y  e.  D  |->  R ) B )
1716nfeq1 2232 . . . 4  |-  F/ y ( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S
182, 3, 4, 7, 8, 12, 13, 17ovmpt2df 5711 . . 3  |-  ( ph  ->  ( ( x  e.  C ,  y  e.  D  |->  R )  =  ( x  e.  C ,  y  e.  D  |->  R )  ->  ( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S ) )
191, 18mpd 13 . 2  |-  ( ph  ->  ( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S )
20 oveq 5597 . . 3  |-  ( F  =  ( x  e.  C ,  y  e.  D  |->  R )  -> 
( A F B )  =  ( A ( x  e.  C ,  y  e.  D  |->  R ) B ) )
2120eqeq1d 2091 . 2  |-  ( F  =  ( x  e.  C ,  y  e.  D  |->  R )  -> 
( ( A F B )  =  S  <-> 
( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S ) )
2219, 21syl5ibrcom 155 1  |-  ( ph  ->  ( F  =  ( x  e.  C , 
y  e.  D  |->  R )  ->  ( A F B )  =  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434  (class class class)co 5591    |-> cmpt2 5593
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000  ax-setind 4316
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-v 2614  df-sbc 2827  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-id 4084  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-iota 4934  df-fun 4971  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596
This theorem is referenced by: (None)
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