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Theorem dmmptif 44524
Description: Domain of the mapping operation. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
Hypotheses
Ref Expression
dmmptif.1 𝑥𝐴
dmmptif.2 𝐵 ∈ V
dmmptif.3 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
dmmptif dom 𝐹 = 𝐴

Proof of Theorem dmmptif
StepHypRef Expression
1 dmmptif.1 . . 3 𝑥𝐴
2 dmmptif.2 . . 3 𝐵 ∈ V
3 dmmptif.3 . . 3 𝐹 = (𝑥𝐴𝐵)
41, 2, 3fnmptif 44523 . 2 𝐹 Fn 𝐴
5 fndm 6645 . 2 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
64, 5ax-mp 5 1 dom 𝐹 = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098  wnfc 2877  Vcvv 3468  cmpt 5224  dom cdm 5669   Fn wfn 6531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-fun 6538  df-fn 6539
This theorem is referenced by:  adddmmbl2  46103  muldmmbl2  46105  smfdivdmmbl2  46110
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