Theorem relssinxpdmrn 38391

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 6216	. . 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 4186	. 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 3896   ⊆ wss 3897   × cxp 5612  dom cdm 5614  ran crn 5615  Rel wrel 5619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-xp 5620  df-rel 5621  df-cnv 5622  df-dm 5624  df-rn 5625

This theorem is referenced by:  cnvref4  38392