Theorem dmmptssf 43692

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 6225	. 2 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
3		dmmptssf.1	. . 3 ⊢ Ⅎ𝑥𝐴
4	3	ssrab2f 43563	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ⊆ 𝐴
5	2, 4	eqsstri 4009	1 ⊢ dom 𝐹 ⊆ 𝐴

Syntax hints:   = wceq 1541   ∈ wcel 2106  Ⅎwnfc 2882  {crab 3429  Vcvv 3470   ⊆ wss 3941   ↦ cmpt 5221  dom cdm 5666

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rab 3430  df-v 3472  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4520  df-sn 4620  df-pr 4622  df-op 4626  df-br 5139  df-opab 5201  df-mpt 5222  df-xp 5672  df-rel 5673  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679

This theorem is referenced by:  rn1st  43737  limsupequzmpt2  44193  liminfequzmpt2  44266