MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovmpt2dx Unicode version

Theorem ovmpt2dx 5974
Description: Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 29-Dec-2014.)
Hypotheses
Ref Expression
ovmpt2dx.1  |-  ( ph  ->  F  =  ( x  e.  C ,  y  e.  D  |->  R ) )
ovmpt2dx.2  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  R  =  S )
ovmpt2dx.3  |-  ( (
ph  /\  x  =  A )  ->  D  =  L )
ovmpt2dx.4  |-  ( ph  ->  A  e.  C )
ovmpt2dx.5  |-  ( ph  ->  B  e.  L )
ovmpt2dx.6  |-  ( ph  ->  S  e.  X )
Assertion
Ref Expression
ovmpt2dx  |-  ( ph  ->  ( A F B )  =  S )
Distinct variable groups:    x, y, A    y, B    y, A    x, B    x, S, y    ph, x, y
Allowed substitution hints:    C( x, y)    D( x, y)    R( x, y)    F( x, y)    L( x, y)    X( x, y)

Proof of Theorem ovmpt2dx
StepHypRef Expression
1 ovmpt2dx.1 . 2  |-  ( ph  ->  F  =  ( x  e.  C ,  y  e.  D  |->  R ) )
2 ovmpt2dx.2 . 2  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  R  =  S )
3 ovmpt2dx.3 . 2  |-  ( (
ph  /\  x  =  A )  ->  D  =  L )
4 ovmpt2dx.4 . 2  |-  ( ph  ->  A  e.  C )
5 ovmpt2dx.5 . 2  |-  ( ph  ->  B  e.  L )
6 ovmpt2dx.6 . 2  |-  ( ph  ->  S  e.  X )
7 nfv 1605 . 2  |-  F/ x ph
8 nfv 1605 . 2  |-  F/ y
ph
9 nfcv 2419 . 2  |-  F/_ y A
10 nfcv 2419 . 2  |-  F/_ x B
11 nfcv 2419 . 2  |-  F/_ x S
12 nfcv 2419 . 2  |-  F/_ y S
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12ovmpt2dxf 5973 1  |-  ( ph  ->  ( A F B )  =  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684  (class class class)co 5858    e. cmpt2 5860
This theorem is referenced by:  ovmpt2d  5975  ovmpt2x  5976  dpjfval  15290  fgval  17565  om1val  18528  pi1val  18535  dvfval  19247  dvnfval  19271  taylfval  19738
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
  Copyright terms: Public domain W3C validator