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Theorem ovmpt2dv2 6421
Description: Alternate deduction version of ovmpt2 6423, 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 2457 . . . . 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 6349 . . . 4  |-  F/_ x
( x  e.  C ,  y  e.  D  |->  R )
9 nfcv 2605 . . . . . 6  |-  F/_ x A
10 nfcv 2605 . . . . . 6  |-  F/_ x B
119, 8, 10nfov 6307 . . . . 5  |-  F/_ x
( A ( x  e.  C ,  y  e.  D  |->  R ) B )
1211nfeq1 2620 . . . 4  |-  F/ x
( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S
13 nfmpt22 6350 . . . 4  |-  F/_ y
( x  e.  C ,  y  e.  D  |->  R )
14 nfcv 2605 . . . . . 6  |-  F/_ y A
15 nfcv 2605 . . . . . 6  |-  F/_ y B
1614, 13, 15nfov 6307 . . . . 5  |-  F/_ y
( A ( x  e.  C ,  y  e.  D  |->  R ) B )
1716nfeq1 2620 . . . 4  |-  F/ y ( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S
182, 3, 4, 7, 8, 12, 13, 17ovmpt2df 6419 . . 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 6287 . . 3  |-  ( F  =  ( x  e.  C ,  y  e.  D  |->  R )  -> 
( A F B )  =  ( A ( x  e.  C ,  y  e.  D  |->  R ) B ) )
2120eqeq1d 2445 . 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 1383    e. wcel 1804  (class class class)co 6281    |-> cmpt2 6283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-iota 5541  df-fun 5580  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286
This theorem is referenced by:  coaval  15269  xpcco  15326  marrepval  18937  marrepeval  18938  marepveval  18943  submaval  18956  submaeval  18957  minmar1val  19023  minmar1eval  19024  nbgraop  24295  isuvtx  24360
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