Theorem dmmptif 45710

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 45709	. 2 ⊢ 𝐹 Fn 𝐴
5		fndm 6588	. 2 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
6	4, 5	ax-mp 5	1 ⊢ dom 𝐹 = 𝐴

Syntax hints:   = wceq 1547   ∈ wcel 2119  Ⅎwnfc 2886  Vcvv 3431   ↦ cmpt 5153  dom cdm 5618   Fn wfn 6480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-fun 6487  df-fn 6488

This theorem is referenced by:  adddmmbl2  47277  muldmmbl2  47279  smfdivdmmbl2  47284