Theorem dmiin 5925

Description: Domain of an intersection. (Contributed by FL, 15-Oct-2012.)

Assertion

Ref	Expression
dmiin	⊢ dom ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ ∩ 𝑥 ∈ 𝐴 dom 𝐵

Proof of Theorem dmiin

Step	Hyp	Ref	Expression
1		nfii1 4983	. . . 4 ⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 𝐵
2	1	nfdm 5923	. . 3 ⊢ Ⅎ𝑥dom ∩ 𝑥 ∈ 𝐴 𝐵
3	2	ssiinf 5009	. 2 ⊢ (dom ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ ∩ 𝑥 ∈ 𝐴 dom 𝐵 ↔ ∀𝑥 ∈ 𝐴 dom ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ dom 𝐵)
4		iinss2 5012	. . 3 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵)
5		dmss 5874	. . 3 ⊢ (∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵 → dom ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ dom 𝐵)
6	4, 5	syl 17	. 2 ⊢ (𝑥 ∈ 𝐴 → dom ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ dom 𝐵)
7	3, 6	mprgbir 3082	1 ⊢ dom ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ ∩ 𝑥 ∈ 𝐴 dom 𝐵

Syntax hints:   ∈ wcel 2141   ⊆ wss 3902  ∩ ciin 4947  dom cdm 5643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3076  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-iin 4949  df-br 5098  df-dm 5653

This theorem is referenced by: (None)