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Theorem ovmpt2df 5979
Description: Alternate deduction version of ovmpt2 5983, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
Hypotheses
Ref Expression
ovmpt2df.1  |-  ( ph  ->  A  e.  C )
ovmpt2df.2  |-  ( (
ph  /\  x  =  A )  ->  B  e.  D )
ovmpt2df.3  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  R  e.  V )
ovmpt2df.4  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( ( A F B )  =  R  ->  ps ) )
ovmpt2df.5  |-  F/_ x F
ovmpt2df.6  |-  F/ x ps
ovmpt2df.7  |-  F/_ y F
ovmpt2df.8  |-  F/ y ps
Assertion
Ref Expression
ovmpt2df  |-  ( ph  ->  ( F  =  ( x  e.  C , 
y  e.  D  |->  R )  ->  ps )
)
Distinct variable groups:    x, y, A    y, B    ph, x, y
Allowed substitution hints:    ps( x, y)    B( x)    C( x, y)    D( x, y)    R( x, y)    F( x, y)    V( x, y)

Proof of Theorem ovmpt2df
StepHypRef Expression
1 ovmpt2df.1 . . . 4  |-  ( ph  ->  A  e.  C )
2 elex 2796 . . . 4  |-  ( A  e.  C  ->  A  e.  _V )
31, 2syl 15 . . 3  |-  ( ph  ->  A  e.  _V )
4 isset 2792 . . 3  |-  ( A  e.  _V  <->  E. x  x  =  A )
53, 4sylib 188 . 2  |-  ( ph  ->  E. x  x  =  A )
6 nfv 1605 . . 3  |-  F/ x ph
7 ovmpt2df.5 . . . . 5  |-  F/_ x F
8 nfmpt21 5914 . . . . 5  |-  F/_ x
( x  e.  C ,  y  e.  D  |->  R )
97, 8nfeq 2426 . . . 4  |-  F/ x  F  =  ( x  e.  C ,  y  e.  D  |->  R )
10 ovmpt2df.6 . . . 4  |-  F/ x ps
119, 10nfim 1769 . . 3  |-  F/ x
( F  =  ( x  e.  C , 
y  e.  D  |->  R )  ->  ps )
12 ovmpt2df.2 . . . . . . 7  |-  ( (
ph  /\  x  =  A )  ->  B  e.  D )
13 elex 2796 . . . . . . 7  |-  ( B  e.  D  ->  B  e.  _V )
1412, 13syl 15 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  B  e.  _V )
15 isset 2792 . . . . . 6  |-  ( B  e.  _V  <->  E. y 
y  =  B )
1614, 15sylib 188 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  E. y 
y  =  B )
17 nfv 1605 . . . . . 6  |-  F/ y ( ph  /\  x  =  A )
18 ovmpt2df.7 . . . . . . . 8  |-  F/_ y F
19 nfmpt22 5915 . . . . . . . 8  |-  F/_ y
( x  e.  C ,  y  e.  D  |->  R )
2018, 19nfeq 2426 . . . . . . 7  |-  F/ y  F  =  ( x  e.  C ,  y  e.  D  |->  R )
21 ovmpt2df.8 . . . . . . 7  |-  F/ y ps
2220, 21nfim 1769 . . . . . 6  |-  F/ y ( F  =  ( x  e.  C , 
y  e.  D  |->  R )  ->  ps )
23 oveq 5864 . . . . . . . 8  |-  ( F  =  ( x  e.  C ,  y  e.  D  |->  R )  -> 
( A F B )  =  ( A ( x  e.  C ,  y  e.  D  |->  R ) B ) )
24 simprl 732 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  x  =  A )
25 simprr 733 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
y  =  B )
2624, 25oveq12d 5876 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  ( A ( x  e.  C ,  y  e.  D  |->  R ) B ) )
271adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  A  e.  C )
2824, 27eqeltrd 2357 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  x  e.  C )
2912adantrr 697 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  B  e.  D )
3025, 29eqeltrd 2357 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
y  e.  D )
31 ovmpt2df.3 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  R  e.  V )
32 eqid 2283 . . . . . . . . . . . . 13  |-  ( x  e.  C ,  y  e.  D  |->  R )  =  ( x  e.  C ,  y  e.  D  |->  R )
3332ovmpt4g 5970 . . . . . . . . . . . 12  |-  ( ( x  e.  C  /\  y  e.  D  /\  R  e.  V )  ->  ( x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  R )
3428, 30, 31, 33syl3anc 1182 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  R )
3526, 34eqtr3d 2317 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  R )
3635eqeq2d 2294 . . . . . . . . 9  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( ( A F B )  =  ( A ( x  e.  C ,  y  e.  D  |->  R ) B )  <->  ( A F B )  =  R ) )
37 ovmpt2df.4 . . . . . . . . 9  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( ( A F B )  =  R  ->  ps ) )
3836, 37sylbid 206 . . . . . . . 8  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( ( A F B )  =  ( A ( x  e.  C ,  y  e.  D  |->  R ) B )  ->  ps )
)
3923, 38syl5 28 . . . . . . 7  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( F  =  ( x  e.  C , 
y  e.  D  |->  R )  ->  ps )
)
4039expr 598 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  (
y  =  B  -> 
( F  =  ( x  e.  C , 
y  e.  D  |->  R )  ->  ps )
) )
4117, 22, 40exlimd 1803 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  ( E. y  y  =  B  ->  ( F  =  ( x  e.  C ,  y  e.  D  |->  R )  ->  ps ) ) )
4216, 41mpd 14 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  ( F  =  ( x  e.  C ,  y  e.  D  |->  R )  ->  ps ) )
4342ex 423 . . 3  |-  ( ph  ->  ( x  =  A  ->  ( F  =  ( x  e.  C ,  y  e.  D  |->  R )  ->  ps ) ) )
446, 11, 43exlimd 1803 . 2  |-  ( ph  ->  ( E. x  x  =  A  ->  ( F  =  ( x  e.  C ,  y  e.  D  |->  R )  ->  ps ) ) )
455, 44mpd 14 1  |-  ( ph  ->  ( F  =  ( x  e.  C , 
y  e.  D  |->  R )  ->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528   F/wnf 1531    = wceq 1623    e. wcel 1684   F/_wnfc 2406   _Vcvv 2788  (class class class)co 5858    e. cmpt2 5860
This theorem is referenced by:  ovmpt2dv  5980  ovmpt2dv2  5981
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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