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Theorem dmmptif 45178
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 45177 . 2 𝐹 Fn 𝐴
5 fndm 6684 . 2 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
64, 5ax-mp 5 1 dom 𝐹 = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2108  wnfc 2893  Vcvv 3488  cmpt 5249  dom cdm 5700   Fn wfn 6570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-fun 6577  df-fn 6578
This theorem is referenced by:  adddmmbl2  46757  muldmmbl2  46759  smfdivdmmbl2  46764
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