Theorem relssinxpdmrn 38853

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 6258	. . 3 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅))
2	1	biantrud 539	. 2 ⊢ (Rel 𝑅 → (𝑅 ⊆ 𝑆 ↔ (𝑅 ⊆ 𝑆 ∧ 𝑅 ⊆ (dom 𝑅 × ran 𝑅))))
3		ssin 4192	. 2 ⊢ ((𝑅 ⊆ 𝑆 ∧ 𝑅 ⊆ (dom 𝑅 × ran 𝑅)) ↔ 𝑅 ⊆ (𝑆 ∩ (dom 𝑅 × ran 𝑅)))
4	2, 3	bitr2di 290	1 ⊢ (Rel 𝑅 → (𝑅 ⊆ (𝑆 ∩ (dom 𝑅 × ran 𝑅)) ↔ 𝑅 ⊆ 𝑆))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∩ cin 3905   ⊆ wss 3906   × cxp 5647  dom cdm 5649  ran crn 5650  Rel wrel 5654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-xp 5655  df-rel 5656  df-cnv 5657  df-dm 5659  df-rn 5660

This theorem is referenced by:  cnvref4  38854