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Theorem dmmpt2ga 6845
Description: Domain of an operation given by the "maps to" notation, closed form of dmmpt2 6843. (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 6841 . 2  |-  ( A. x  e.  A  A. y  e.  B  C  e.  V  ->  F  Fn  ( A  X.  B
) )
3 fndm 5662 . 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 1398    e. wcel 1823   A.wral 2804    X. cxp 4986   dom cdm 4988    Fn wfn 5565    |-> cmpt2 6272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774
This theorem is referenced by:  dmmpt2g  6846  mpt2curryd  6990  mamudm  19057  mavmuldm  19219  mavmul0g  19222
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