Theorem relssinxpdmrn 38629

Description: Subset of restriction, special case. (Contributed by Peter Mazsa, 10-Apr-2023.)

Assertion

Ref	Expression
relssinxpdmrn	⊢ (Rel 𝑅 → (𝑅 ⊆ (𝑆 ∩ (dom 𝑅 × ran 𝑅)) ↔ 𝑅 ⊆ 𝑆))

Proof of Theorem relssinxpdmrn

Step	Hyp	Ref	Expression
1		relssdmrn 6237	. . 3 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅))
2	1	biantrud 531	. 2 ⊢ (Rel 𝑅 → (𝑅 ⊆ 𝑆 ↔ (𝑅 ⊆ 𝑆 ∧ 𝑅 ⊆ (dom 𝑅 × ran 𝑅))))
3		ssin 4193	. 2 ⊢ ((𝑅 ⊆ 𝑆 ∧ 𝑅 ⊆ (dom 𝑅 × ran 𝑅)) ↔ 𝑅 ⊆ (𝑆 ∩ (dom 𝑅 × ran 𝑅)))
4	2, 3	bitr2di 288	1 ⊢ (Rel 𝑅 → (𝑅 ⊆ (𝑆 ∩ (dom 𝑅 × ran 𝑅)) ↔ 𝑅 ⊆ 𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∩ cin 3902   ⊆ wss 3903   × cxp 5632  dom cdm 5634  ran crn 5635  Rel wrel 5639

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  ax-sep 5245  ax-pr 5381

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-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5640  df-rel 5641  df-cnv 5642  df-dm 5644  df-rn 5645

This theorem is referenced by:  cnvref4  38630