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Theorem ovmpodf 5854
Description: Alternate deduction version of ovmpo 5858, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
Hypotheses
Ref Expression
ovmpodf.1  |-  ( ph  ->  A  e.  C )
ovmpodf.2  |-  ( (
ph  /\  x  =  A )  ->  B  e.  D )
ovmpodf.3  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  R  e.  V )
ovmpodf.4  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( ( A F B )  =  R  ->  ps ) )
ovmpodf.5  |-  F/_ x F
ovmpodf.6  |-  F/ x ps
ovmpodf.7  |-  F/_ y F
ovmpodf.8  |-  F/ y ps
Assertion
Ref Expression
ovmpodf  |-  ( 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 ovmpodf
StepHypRef Expression
1 nfv 1489 . 2  |-  F/ x ph
2 ovmpodf.5 . . . 4  |-  F/_ x F
3 nfmpo1 5790 . . . 4  |-  F/_ x
( x  e.  C ,  y  e.  D  |->  R )
42, 3nfeq 2261 . . 3  |-  F/ x  F  =  ( x  e.  C ,  y  e.  D  |->  R )
5 ovmpodf.6 . . 3  |-  F/ x ps
64, 5nfim 1532 . 2  |-  F/ x
( F  =  ( x  e.  C , 
y  e.  D  |->  R )  ->  ps )
7 ovmpodf.1 . . . 4  |-  ( ph  ->  A  e.  C )
8 elex 2666 . . . 4  |-  ( A  e.  C  ->  A  e.  _V )
97, 8syl 14 . . 3  |-  ( ph  ->  A  e.  _V )
10 isset 2661 . . 3  |-  ( A  e.  _V  <->  E. x  x  =  A )
119, 10sylib 121 . 2  |-  ( ph  ->  E. x  x  =  A )
12 ovmpodf.2 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  B  e.  D )
13 elex 2666 . . . . 5  |-  ( B  e.  D  ->  B  e.  _V )
1412, 13syl 14 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  B  e.  _V )
15 isset 2661 . . . 4  |-  ( B  e.  _V  <->  E. y 
y  =  B )
1614, 15sylib 121 . . 3  |-  ( (
ph  /\  x  =  A )  ->  E. y 
y  =  B )
17 nfv 1489 . . . 4  |-  F/ y ( ph  /\  x  =  A )
18 ovmpodf.7 . . . . . 6  |-  F/_ y F
19 nfmpo2 5791 . . . . . 6  |-  F/_ y
( x  e.  C ,  y  e.  D  |->  R )
2018, 19nfeq 2261 . . . . 5  |-  F/ y  F  =  ( x  e.  C ,  y  e.  D  |->  R )
21 ovmpodf.8 . . . . 5  |-  F/ y ps
2220, 21nfim 1532 . . . 4  |-  F/ y ( F  =  ( x  e.  C , 
y  e.  D  |->  R )  ->  ps )
23 oveq 5732 . . . . . 6  |-  ( F  =  ( x  e.  C ,  y  e.  D  |->  R )  -> 
( A F B )  =  ( A ( x  e.  C ,  y  e.  D  |->  R ) B ) )
24 simprl 503 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  x  =  A )
25 simprr 504 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
y  =  B )
2624, 25oveq12d 5744 . . . . . . . . 9  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  ( A ( x  e.  C ,  y  e.  D  |->  R ) B ) )
277adantr 272 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  A  e.  C )
2824, 27eqeltrd 2189 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  x  e.  C )
2912adantrr 468 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  B  e.  D )
3025, 29eqeltrd 2189 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
y  e.  D )
31 ovmpodf.3 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  R  e.  V )
32 eqid 2113 . . . . . . . . . . 11  |-  ( x  e.  C ,  y  e.  D  |->  R )  =  ( x  e.  C ,  y  e.  D  |->  R )
3332ovmpt4g 5845 . . . . . . . . . 10  |-  ( ( x  e.  C  /\  y  e.  D  /\  R  e.  V )  ->  ( x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  R )
3428, 30, 31, 33syl3anc 1197 . . . . . . . . 9  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  R )
3526, 34eqtr3d 2147 . . . . . . . 8  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  R )
3635eqeq2d 2124 . . . . . . 7  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( ( A F B )  =  ( A ( x  e.  C ,  y  e.  D  |->  R ) B )  <->  ( A F B )  =  R ) )
37 ovmpodf.4 . . . . . . 7  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( ( A F B )  =  R  ->  ps ) )
3836, 37sylbid 149 . . . . . 6  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( ( A F B )  =  ( A ( x  e.  C ,  y  e.  D  |->  R ) B )  ->  ps )
)
3923, 38syl5 32 . . . . 5  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( F  =  ( x  e.  C , 
y  e.  D  |->  R )  ->  ps )
)
4039expr 370 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  (
y  =  B  -> 
( F  =  ( x  e.  C , 
y  e.  D  |->  R )  ->  ps )
) )
4117, 22, 40exlimd 1557 . . 3  |-  ( (
ph  /\  x  =  A )  ->  ( E. y  y  =  B  ->  ( F  =  ( x  e.  C ,  y  e.  D  |->  R )  ->  ps ) ) )
4216, 41mpd 13 . 2  |-  ( (
ph  /\  x  =  A )  ->  ( F  =  ( x  e.  C ,  y  e.  D  |->  R )  ->  ps ) )
431, 6, 11, 42exlimdd 1824 1  |-  ( ph  ->  ( F  =  ( x  e.  C , 
y  e.  D  |->  R )  ->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1312   F/wnf 1417   E.wex 1449    e. wcel 1461   F/_wnfc 2240   _Vcvv 2655  (class class class)co 5726    e. cmpo 5728
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-setind 4410
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-v 2657  df-sbc 2877  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-iota 5044  df-fun 5081  df-fv 5087  df-ov 5729  df-oprab 5730  df-mpo 5731
This theorem is referenced by:  ovmpodv  5855  ovmpodv2  5856
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