Theorem dmmptif 45111

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 45110	. 2 ⊢ 𝐹 Fn 𝐴
5		fndm 6681	. 2 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
6	4, 5	ax-mp 5	1 ⊢ dom 𝐹 = 𝐴

Syntax hints:   = wceq 1537   ∈ wcel 2103  Ⅎwnfc 2888  Vcvv 3482   ↦ cmpt 5252  dom cdm 5699   Fn wfn 6567

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 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450

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 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-fun 6574  df-fn 6575

This theorem is referenced by:  adddmmbl2  46690  muldmmbl2  46692  smfdivdmmbl2  46697