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Theorem ovmpt2dv2 5981
Description: Alternate deduction version of ovmpt2 5983, 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 2284 . . 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 2294 . . . . 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 198 . . . 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 5914 . . . 4  |-  F/_ x
( x  e.  C ,  y  e.  D  |->  R )
9 nfcv 2419 . . . . . 6  |-  F/_ x A
10 nfcv 2419 . . . . . 6  |-  F/_ x B
119, 8, 10nfov 5881 . . . . 5  |-  F/_ x
( A ( x  e.  C ,  y  e.  D  |->  R ) B )
1211nfeq1 2428 . . . 4  |-  F/ x
( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S
13 nfmpt22 5915 . . . 4  |-  F/_ y
( x  e.  C ,  y  e.  D  |->  R )
14 nfcv 2419 . . . . . 6  |-  F/_ y A
15 nfcv 2419 . . . . . 6  |-  F/_ y B
1614, 13, 15nfov 5881 . . . . 5  |-  F/_ y
( A ( x  e.  C ,  y  e.  D  |->  R ) B )
1716nfeq1 2428 . . . 4  |-  F/ y ( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S
182, 3, 4, 7, 8, 12, 13, 17ovmpt2df 5979 . . 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 14 . 2  |-  ( ph  ->  ( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S )
20 oveq 5864 . . 3  |-  ( F  =  ( x  e.  C ,  y  e.  D  |->  R )  -> 
( A F B )  =  ( A ( x  e.  C ,  y  e.  D  |->  R ) B ) )
2120eqeq1d 2291 . 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 213 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 358    = wceq 1623    e. wcel 1684  (class class class)co 5858    e. cmpt2 5860
This theorem is referenced by:  coaval  13900  xpcco  13957  nbgraop  28140  isuvtx  28160
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863
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