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Theorem ovmpt2dv2 6430
Description: Alternate deduction version of ovmpt2 6432, 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 2468 . . 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 2481 . . . . 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 6358 . . . 4  |-  F/_ x
( x  e.  C ,  y  e.  D  |->  R )
9 nfcv 2629 . . . . . 6  |-  F/_ x A
10 nfcv 2629 . . . . . 6  |-  F/_ x B
119, 8, 10nfov 6317 . . . . 5  |-  F/_ x
( A ( x  e.  C ,  y  e.  D  |->  R ) B )
1211nfeq1 2644 . . . 4  |-  F/ x
( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S
13 nfmpt22 6359 . . . 4  |-  F/_ y
( x  e.  C ,  y  e.  D  |->  R )
14 nfcv 2629 . . . . . 6  |-  F/_ y A
15 nfcv 2629 . . . . . 6  |-  F/_ y B
1614, 13, 15nfov 6317 . . . . 5  |-  F/_ y
( A ( x  e.  C ,  y  e.  D  |->  R ) B )
1716nfeq1 2644 . . . 4  |-  F/ y ( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  S
182, 3, 4, 7, 8, 12, 13, 17ovmpt2df 6428 . . 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 6300 . . 3  |-  ( F  =  ( x  e.  C ,  y  e.  D  |->  R )  -> 
( A F B )  =  ( A ( x  e.  C ,  y  e.  D  |->  R ) B ) )
2120eqeq1d 2469 . 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 1379    e. wcel 1767  (class class class)co 6294    |-> cmpt2 6296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-br 4453  df-opab 4511  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-iota 5556  df-fun 5595  df-fv 5601  df-ov 6297  df-oprab 6298  df-mpt2 6299
This theorem is referenced by:  coaval  15265  xpcco  15322  marrepval  18910  marrepeval  18911  marepveval  18916  submaval  18929  submaeval  18930  minmar1val  18996  minmar1eval  18997  nbgraop  24214  isuvtx  24279
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