Theorem relssinxpdmrn 38350

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 6288	. . 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 4239	. 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 3950   ⊆ wss 3951   × cxp 5683  dom cdm 5685  ran crn 5686  Rel wrel 5690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696

This theorem is referenced by:  cnvref4  38351