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Theorem dmmpti 5129
Description: Domain of an ordered-pair class abstraction that specifies a function. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypotheses
Ref Expression
fnmpti.1  |-  B  e. 
_V
fnmpti.2  |-  F  =  ( x  e.  A  |->  B )
Assertion
Ref Expression
dmmpti  |-  dom  F  =  A
Distinct variable group:    x, A
Allowed substitution hints:    B( x)    F( x)

Proof of Theorem dmmpti
StepHypRef Expression
1 fnmpti.1 . . 3  |-  B  e. 
_V
2 fnmpti.2 . . 3  |-  F  =  ( x  e.  A  |->  B )
31, 2fnmpti 5128 . 2  |-  F  Fn  A
4 fndm 5099 . 2  |-  ( F  Fn  A  ->  dom  F  =  A )
53, 4ax-mp 7 1  |-  dom  F  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1289    e. wcel 1438   _Vcvv 2619    |-> cmpt 3891   dom cdm 4428    Fn wfn 4997
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-fun 5004  df-fn 5005
This theorem is referenced by:  brtpos2  5998
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