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Theorem ovmpt2dv2 6224
Description: Alternate deduction version of ovmpt2 6226, 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 2444 . . 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 2454 . . . . 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 207 . . . 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 6153 . . . 4  |-  F/_ x
( x  e.  C ,  y  e.  D  |->  R )
9 nfcv 2579 . . . . . 6  |-  F/_ x A
10 nfcv 2579 . . . . . 6  |-  F/_ x B
119, 8, 10nfov 6114 . . . . 5  |-  F/_ x
( A ( x  e.  C ,  y  e.  D  |->  R ) B )
1211nfeq1 2588 . . . 4  |-  F/ x
( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S
13 nfmpt22 6154 . . . 4  |-  F/_ y
( x  e.  C ,  y  e.  D  |->  R )
14 nfcv 2579 . . . . . 6  |-  F/_ y A
15 nfcv 2579 . . . . . 6  |-  F/_ y B
1614, 13, 15nfov 6114 . . . . 5  |-  F/_ y
( A ( x  e.  C ,  y  e.  D  |->  R ) B )
1716nfeq1 2588 . . . 4  |-  F/ y ( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S
182, 3, 4, 7, 8, 12, 13, 17ovmpt2df 6222 . . 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 15 . 2  |-  ( ph  ->  ( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S )
20 oveq 6097 . . 3  |-  ( F  =  ( x  e.  C ,  y  e.  D  |->  R )  -> 
( A F B )  =  ( A ( x  e.  C ,  y  e.  D  |->  R ) B ) )
2120eqeq1d 2451 . 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 222 1  |-  ( ph  ->  ( F  =  ( x  e.  C , 
y  e.  D  |->  R )  ->  ( A F B )  =  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756  (class class class)co 6091    e. cmpt2 6093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096
This theorem is referenced by:  coaval  14936  xpcco  14993  marrepval  18373  marrepeval  18374  marepveval  18379  submaval  18392  submaeval  18393  minmar1val  18454  minmar1eval  18455  nbgraop  23335  isuvtx  23396
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