Theorem dmmptssf 44506

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 6233	. 2 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
3		dmmptssf.1	. . 3 ⊢ Ⅎ𝑥𝐴
4	3	ssrab2f 44381	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ⊆ 𝐴
5	2, 4	eqsstri 4011	1 ⊢ dom 𝐹 ⊆ 𝐴

Syntax hints:   = wceq 1533   ∈ wcel 2098  Ⅎwnfc 2877  {crab 3426  Vcvv 3468   ⊆ wss 3943   ↦ cmpt 5224  dom cdm 5669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-mpt 5225  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682

This theorem is referenced by:  rn1st  44550  limsupequzmpt2  45006  liminfequzmpt2  45079