Theorem ssdmral 38978

Description: Subclass of a domain. (Contributed by Peter Mazsa, 15-Sep-2018.)

Assertion

Ref	Expression
ssdmral	⊢ (𝐴 ⊆ dom 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 𝑥𝑅𝑦)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑅,𝑦

Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem ssdmral

Step	Hyp	Ref	Expression
1		dfss3 3934	. 2 ⊢ (𝐴 ⊆ dom 𝑅 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ dom 𝑅)
2		eldmg 5892	. . . 4 ⊢ (𝑥 ∈ V → (𝑥 ∈ dom 𝑅 ↔ ∃𝑦 𝑥𝑅𝑦))
3	2	elv 3467	. . 3 ⊢ (𝑥 ∈ dom 𝑅 ↔ ∃𝑦 𝑥𝑅𝑦)
4	3	ralbii 3118	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ dom 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 𝑥𝑅𝑦)
5	1, 4	bitri 278	1 ⊢ (𝐴 ⊆ dom 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 𝑥𝑅𝑦)

Syntax hints:   ↔ wb 209  ∃wex 1807   ∈ wcel 2150  ∀wral 3086  Vcvv 3462   ⊆ wss 3913   class class class wbr 5114  dom cdm 5665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-ral 3087  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-dm 5675

This theorem is referenced by:  dmsucmap  39067