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Theorem ovmpt2df 6205
Description: Alternate deduction version of ovmpt2 6209, 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 nfv 1629 . 2  |-  F/ x ph
2 ovmpt2df.5 . . . 4  |-  F/_ x F
3 nfmpt21 6140 . . . 4  |-  F/_ x
( x  e.  C ,  y  e.  D  |->  R )
42, 3nfeq 2579 . . 3  |-  F/ x  F  =  ( x  e.  C ,  y  e.  D  |->  R )
5 ovmpt2df.6 . . 3  |-  F/ x ps
64, 5nfim 1832 . 2  |-  F/ x
( F  =  ( x  e.  C , 
y  e.  D  |->  R )  ->  ps )
7 ovmpt2df.1 . . . 4  |-  ( ph  ->  A  e.  C )
8 elex 2964 . . . 4  |-  ( A  e.  C  ->  A  e.  _V )
97, 8syl 16 . . 3  |-  ( ph  ->  A  e.  _V )
10 isset 2960 . . 3  |-  ( A  e.  _V  <->  E. x  x  =  A )
119, 10sylib 189 . 2  |-  ( ph  ->  E. x  x  =  A )
12 ovmpt2df.2 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  B  e.  D )
13 elex 2964 . . . . 5  |-  ( B  e.  D  ->  B  e.  _V )
1412, 13syl 16 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  B  e.  _V )
15 isset 2960 . . . 4  |-  ( B  e.  _V  <->  E. y 
y  =  B )
1614, 15sylib 189 . . 3  |-  ( (
ph  /\  x  =  A )  ->  E. y 
y  =  B )
17 nfv 1629 . . . 4  |-  F/ y ( ph  /\  x  =  A )
18 ovmpt2df.7 . . . . . 6  |-  F/_ y F
19 nfmpt22 6141 . . . . . 6  |-  F/_ y
( x  e.  C ,  y  e.  D  |->  R )
2018, 19nfeq 2579 . . . . 5  |-  F/ y  F  =  ( x  e.  C ,  y  e.  D  |->  R )
21 ovmpt2df.8 . . . . 5  |-  F/ y ps
2220, 21nfim 1832 . . . 4  |-  F/ y ( F  =  ( x  e.  C , 
y  e.  D  |->  R )  ->  ps )
23 oveq 6087 . . . . . 6  |-  ( F  =  ( x  e.  C ,  y  e.  D  |->  R )  -> 
( A F B )  =  ( A ( x  e.  C ,  y  e.  D  |->  R ) B ) )
24 simprl 733 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  x  =  A )
25 simprr 734 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
y  =  B )
2624, 25oveq12d 6099 . . . . . . . . 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 452 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  A  e.  C )
2824, 27eqeltrd 2510 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  x  e.  C )
2912adantrr 698 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  B  e.  D )
3025, 29eqeltrd 2510 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
y  e.  D )
31 ovmpt2df.3 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  R  e.  V )
32 eqid 2436 . . . . . . . . . . 11  |-  ( x  e.  C ,  y  e.  D  |->  R )  =  ( x  e.  C ,  y  e.  D  |->  R )
3332ovmpt4g 6196 . . . . . . . . . 10  |-  ( ( x  e.  C  /\  y  e.  D  /\  R  e.  V )  ->  ( x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  R )
3428, 30, 31, 33syl3anc 1184 . . . . . . . . 9  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  R )
3526, 34eqtr3d 2470 . . . . . . . 8  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  R )
3635eqeq2d 2447 . . . . . . 7  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( ( A F B )  =  ( A ( x  e.  C ,  y  e.  D  |->  R ) B )  <->  ( A F B )  =  R ) )
37 ovmpt2df.4 . . . . . . 7  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( ( A F B )  =  R  ->  ps ) )
3836, 37sylbid 207 . . . . . 6  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( ( A F B )  =  ( A ( x  e.  C ,  y  e.  D  |->  R ) B )  ->  ps )
)
3923, 38syl5 30 . . . . 5  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( F  =  ( x  e.  C , 
y  e.  D  |->  R )  ->  ps )
)
4039expr 599 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  (
y  =  B  -> 
( F  =  ( x  e.  C , 
y  e.  D  |->  R )  ->  ps )
) )
4117, 22, 40exlimd 1824 . . 3  |-  ( (
ph  /\  x  =  A )  ->  ( E. y  y  =  B  ->  ( F  =  ( x  e.  C ,  y  e.  D  |->  R )  ->  ps ) ) )
4216, 41mpd 15 . 2  |-  ( (
ph  /\  x  =  A )  ->  ( F  =  ( x  e.  C ,  y  e.  D  |->  R )  ->  ps ) )
431, 6, 11, 42exlimdd 1912 1  |-  ( ph  ->  ( F  =  ( x  e.  C , 
y  e.  D  |->  R )  ->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1550   F/wnf 1553    = wceq 1652    e. wcel 1725   F/_wnfc 2559   _Vcvv 2956  (class class class)co 6081    e. cmpt2 6083
This theorem is referenced by:  ovmpt2dv  6206  ovmpt2dv2  6207
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086
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