MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovmpt2dv Structured version   Unicode version

Theorem ovmpt2dv 6228
Description: Alternate deduction version of ovmpt2 6231, 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 ) )
Assertion
Ref Expression
ovmpt2dv  |-  ( ph  ->  ( F  =  ( x  e.  C , 
y  e.  D  |->  R )  ->  ps )
)
Distinct variable groups:    x, y, A    y, B    ph, x, y   
x, F, y    ps, x, y
Allowed substitution hints:    B( x)    C( x, y)    D( x, y)    R( x, y)    V( x, y)

Proof of Theorem ovmpt2dv
StepHypRef Expression
1 ovmpt2df.1 . 2  |-  ( ph  ->  A  e.  C )
2 ovmpt2df.2 . 2  |-  ( (
ph  /\  x  =  A )  ->  B  e.  D )
3 ovmpt2df.3 . 2  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  R  e.  V )
4 ovmpt2df.4 . 2  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  -> 
( ( A F B )  =  R  ->  ps ) )
5 nfcv 2584 . 2  |-  F/_ x F
6 nfv 1673 . 2  |-  F/ x ps
7 nfcv 2584 . 2  |-  F/_ y F
8 nfv 1673 . 2  |-  F/ y ps
91, 2, 3, 4, 5, 6, 7, 8ovmpt2df 6227 1  |-  ( ph  ->  ( F  =  ( x  e.  C , 
y  e.  D  |->  R )  ->  ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756  (class class class)co 6096    e. cmpt2 6098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5386  df-fun 5425  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101
This theorem is referenced by:  xpcco  14998  curf12  15042  curf2  15044
  Copyright terms: Public domain W3C validator