Theorem ssdmral 38717

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 3911	. 2 ⊢ (𝐴 ⊆ dom 𝑅 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ dom 𝑅)
2		eldmg 5848	. . . 4 ⊢ (𝑥 ∈ V → (𝑥 ∈ dom 𝑅 ↔ ∃𝑦 𝑥𝑅𝑦))
3	2	elv 3435	. . 3 ⊢ (𝑥 ∈ dom 𝑅 ↔ ∃𝑦 𝑥𝑅𝑦)
4	3	ralbii 3084	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ dom 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 𝑥𝑅𝑦)
5	1, 4	bitri 275	1 ⊢ (𝐴 ⊆ dom 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 𝑥𝑅𝑦)

Syntax hints:   ↔ wb 206  ∃wex 1781   ∈ wcel 2114  ∀wral 3052  Vcvv 3430   ⊆ wss 3890   class class class wbr 5086  dom cdm 5625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-dm 5635

This theorem is referenced by:  dmsucmap  38806