Theorem dmmptssf 45174

Description: The domain of a mapping is a subset of its base class. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

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

Assertion

Ref	Expression
dmmptssf	⊢ dom 𝐹 ⊆ 𝐴

Proof of Theorem dmmptssf

Step	Hyp	Ref	Expression
1		dmmptssf.2	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
2	1	dmmpt 6261	. 2 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
3		dmmptssf.1	. . 3 ⊢ Ⅎ𝑥𝐴
4	3	ssrab2f 45056	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ⊆ 𝐴
5	2, 4	eqsstri 4029	1 ⊢ dom 𝐹 ⊆ 𝐴

Syntax hints:   = wceq 1536   ∈ wcel 2105  Ⅎwnfc 2887  {crab 3432  Vcvv 3477   ⊆ wss 3962   ↦ cmpt 5230  dom cdm 5688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-mpt 5231  df-xp 5694  df-rel 5695  df-cnv 5696  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701

This theorem is referenced by:  rn1st  45218  limsupequzmpt2  45673  liminfequzmpt2  45746