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Theorem dmmptif 44645
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 44644 . 2 𝐹 Fn 𝐴
5 fndm 6660 . 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 2878  Vcvv 3471  cmpt 5233  dom cdm 5680   Fn wfn 6546
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 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-fun 6553  df-fn 6554
This theorem is referenced by:  adddmmbl2  46224  muldmmbl2  46226  smfdivdmmbl2  46231
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