MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmmpt2ga Structured version   Unicode version

Theorem dmmpt2ga 6756
Description: Domain of an operation given by the "maps to" notation, closed form of dmmpt2 6755. (Contributed by Alexander van der Vekens, 10-Feb-2019.)
Hypothesis
Ref Expression
dmmpt2g.f  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
Assertion
Ref Expression
dmmpt2ga  |-  ( A. x  e.  A  A. y  e.  B  C  e.  V  ->  dom  F  =  ( A  X.  B ) )
Distinct variable groups:    x, A, y    x, B, y    x, V, y
Allowed substitution hints:    C( x, y)    F( x, y)

Proof of Theorem dmmpt2ga
StepHypRef Expression
1 dmmpt2g.f . . 3  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
21fnmpt2 6753 . 2  |-  ( A. x  e.  A  A. y  e.  B  C  e.  V  ->  F  Fn  ( A  X.  B
) )
3 fndm 5619 . 2  |-  ( F  Fn  ( A  X.  B )  ->  dom  F  =  ( A  X.  B ) )
42, 3syl 16 1  |-  ( A. x  e.  A  A. y  e.  B  C  e.  V  ->  dom  F  =  ( A  X.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   A.wral 2799    X. cxp 4947   dom cdm 4949    Fn wfn 5522    |-> cmpt2 6203
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-fv 5535  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689
This theorem is referenced by:  dmmpt2g  6757  mpt2curryd  6899  mamudm  18414  mavmuldm  18489  mavmul0g  18492
  Copyright terms: Public domain W3C validator