Theorem dmmptif 45695

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 45694	. 2 ⊢ 𝐹 Fn 𝐴
5		fndm 6601	. 2 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
6	4, 5	ax-mp 5	1 ⊢ dom 𝐹 = 𝐴

Syntax hints:   = wceq 1542   ∈ wcel 2114  Ⅎwnfc 2883  Vcvv 3429   ↦ cmpt 5166  dom cdm 5631   Fn wfn 6493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-fun 6500  df-fn 6501

This theorem is referenced by:  adddmmbl2  47262  muldmmbl2  47264  smfdivdmmbl2  47269