Theorem dmmptssf 45129

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 6266	. 2 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
3		dmmptssf.1	. . 3 ⊢ Ⅎ𝑥𝐴
4	3	ssrab2f 45009	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ⊆ 𝐴
5	2, 4	eqsstri 4043	1 ⊢ dom 𝐹 ⊆ 𝐴

Syntax hints:   = wceq 1537   ∈ wcel 2108  Ⅎwnfc 2893  {crab 3443  Vcvv 3488   ⊆ wss 3976   ↦ cmpt 5249  dom cdm 5695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-mpt 5250  df-xp 5701  df-rel 5702  df-cnv 5703  df-dm 5705  df-rn 5706  df-res 5707  df-ima 5708

This theorem is referenced by:  rn1st  45173  limsupequzmpt2  45629  liminfequzmpt2  45702