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Theorem ovmpt2df 6429
Description: Alternate deduction version of ovmpt2 6433, 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 1683 . 2  |-  F/ x ph
2 ovmpt2df.5 . . . 4  |-  F/_ x F
3 nfmpt21 6359 . . . 4  |-  F/_ x
( x  e.  C ,  y  e.  D  |->  R )
42, 3nfeq 2640 . . 3  |-  F/ x  F  =  ( x  e.  C ,  y  e.  D  |->  R )
5 ovmpt2df.6 . . 3  |-  F/ x ps
64, 5nfim 1867 . 2  |-  F/ x
( F  =  ( x  e.  C , 
y  e.  D  |->  R )  ->  ps )
7 ovmpt2df.1 . . . 4  |-  ( ph  ->  A  e.  C )
8 elex 3127 . . . 4  |-  ( A  e.  C  ->  A  e.  _V )
97, 8syl 16 . . 3  |-  ( ph  ->  A  e.  _V )
10 isset 3122 . . 3  |-  ( A  e.  _V  <->  E. x  x  =  A )
119, 10sylib 196 . 2  |-  ( ph  ->  E. x  x  =  A )
12 ovmpt2df.2 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  B  e.  D )
13 elex 3127 . . . . 5  |-  ( B  e.  D  ->  B  e.  _V )
1412, 13syl 16 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  B  e.  _V )
15 isset 3122 . . . 4  |-  ( B  e.  _V  <->  E. y 
y  =  B )
1614, 15sylib 196 . . 3  |-  ( (
ph  /\  x  =  A )  ->  E. y 
y  =  B )
17 nfv 1683 . . . 4  |-  F/ y ( ph  /\  x  =  A )
18 ovmpt2df.7 . . . . . 6  |-  F/_ y F
19 nfmpt22 6360 . . . . . 6  |-  F/_ y
( x  e.  C ,  y  e.  D  |->  R )
2018, 19nfeq 2640 . . . . 5  |-  F/ y  F  =  ( x  e.  C ,  y  e.  D  |->  R )
21 ovmpt2df.8 . . . . 5  |-  F/ y ps
2220, 21nfim 1867 . . . 4  |-  F/ y ( F  =  ( x  e.  C , 
y  e.  D  |->  R )  ->  ps )
23 oveq 6301 . . . . . 6  |-  ( F  =  ( x  e.  C ,  y  e.  D  |->  R )  -> 
( A F B )  =  ( A ( x  e.  C ,  y  e.  D  |->  R ) B ) )
24 simprl 755 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  x  =  A )
25 simprr 756 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
y  =  B )
2624, 25oveq12d 6313 . . . . . . . . 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 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  A  e.  C )
2824, 27eqeltrd 2555 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  x  e.  C )
2912adantrr 716 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  B  e.  D )
3025, 29eqeltrd 2555 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
y  e.  D )
31 ovmpt2df.3 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  R  e.  V )
32 eqid 2467 . . . . . . . . . . 11  |-  ( x  e.  C ,  y  e.  D  |->  R )  =  ( x  e.  C ,  y  e.  D  |->  R )
3332ovmpt4g 6420 . . . . . . . . . 10  |-  ( ( x  e.  C  /\  y  e.  D  /\  R  e.  V )  ->  ( x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  R )
3428, 30, 31, 33syl3anc 1228 . . . . . . . . 9  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( x ( x  e.  C ,  y  e.  D  |->  R ) y )  =  R )
3526, 34eqtr3d 2510 . . . . . . . 8  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( A ( x  e.  C ,  y  e.  D  |->  R ) B )  =  R )
3635eqeq2d 2481 . . . . . . 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 215 . . . . . 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 615 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  (
y  =  B  -> 
( F  =  ( x  e.  C , 
y  e.  D  |->  R )  ->  ps )
) )
4117, 22, 40exlimd 1861 . . 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 1929 1  |-  ( ph  ->  ( F  =  ( x  e.  C , 
y  e.  D  |->  R )  ->  ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379   E.wex 1596   F/wnf 1599    e. wcel 1767   F/_wnfc 2615   _Vcvv 3118  (class class class)co 6295    |-> cmpt2 6297
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 4574  ax-nul 4582  ax-pr 4692
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 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300
This theorem is referenced by:  ovmpt2dv  6430  ovmpt2dv2  6431
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