Theorem ssdmral 38917

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 5889	. . . 4 ⊢ (𝑥 ∈ V → (𝑥 ∈ dom 𝑅 ↔ ∃𝑦 𝑥𝑅𝑦))
3	2	elv 3468	. . 3 ⊢ (𝑥 ∈ dom 𝑅 ↔ ∃𝑦 𝑥𝑅𝑦)
4	3	ralbii 3117	. 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 1806   ∈ wcel 2149  ∀wral 3085  Vcvv 3463   ⊆ wss 3913   class class class wbr 5113  dom cdm 5662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rab 3424  df-v 3465  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 5114  df-dm 5672

This theorem is referenced by:  dmsucmap  39006