Theorem dmmptif 45232

Description: Domain of the mapping operation. (Contributed by Glauco Siliprandi, 21-Dec-2024.)

Hypotheses

Ref	Expression
dmmptif.1	⊢ Ⅎ𝑥𝐴
dmmptif.2	⊢ 𝐵 ∈ V
dmmptif.3	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)

Assertion

Ref	Expression
dmmptif	⊢ dom 𝐹 = 𝐴

Proof of Theorem dmmptif

Step	Hyp	Ref	Expression
1		dmmptif.1	. . 3 ⊢ Ⅎ𝑥𝐴
2		dmmptif.2	. . 3 ⊢ 𝐵 ∈ V
3		dmmptif.3	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	1, 2, 3	fnmptif 45231	. 2 ⊢ 𝐹 Fn 𝐴
5		fndm 6629	. 2 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
6	4, 5	ax-mp 5	1 ⊢ dom 𝐹 = 𝐴

Syntax hints:   = wceq 1540   ∈ wcel 2109  Ⅎwnfc 2878  Vcvv 3455   ↦ cmpt 5196  dom cdm 5646   Fn wfn 6514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5259  ax-nul 5269  ax-pr 5395

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ral 3047  df-rex 3056  df-rab 3412  df-v 3457  df-dif 3925  df-un 3927  df-ss 3939  df-nul 4305  df-if 4497  df-sn 4598  df-pr 4600  df-op 4604  df-br 5116  df-opab 5178  df-mpt 5197  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-fun 6521  df-fn 6522

This theorem is referenced by:  adddmmbl2  46805  muldmmbl2  46807  smfdivdmmbl2  46812