Theorem dmmptssf 45210

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 6189	. 2 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
3		dmmptssf.1	. . 3 ⊢ Ⅎ𝑥𝐴
4	3	ssrab2f 45095	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ⊆ 𝐴
5	2, 4	eqsstri 3982	1 ⊢ dom 𝐹 ⊆ 𝐴

Syntax hints:   = wceq 1540   ∈ wcel 2109  Ⅎwnfc 2876  {crab 3394  Vcvv 3436   ⊆ wss 3903   ↦ cmpt 5173  dom cdm 5619

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 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-mpt 5174  df-xp 5625  df-rel 5626  df-cnv 5627  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632

This theorem is referenced by:  rn1st  45251  limsupequzmpt2  45699  liminfequzmpt2  45772