Theorem relssinxpdmrn 39084

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 6270	. . 3 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅))
2	1	biantrud 541	. 2 ⊢ (Rel 𝑅 → (𝑅 ⊆ 𝑆 ↔ (𝑅 ⊆ 𝑆 ∧ 𝑅 ⊆ (dom 𝑅 × ran 𝑅))))
3		ssin 4187	. 2 ⊢ ((𝑅 ⊆ 𝑆 ∧ 𝑅 ⊆ (dom 𝑅 × ran 𝑅)) ↔ 𝑅 ⊆ (𝑆 ∩ (dom 𝑅 × ran 𝑅)))
4	2, 3	bitr2di 291	1 ⊢ (Rel 𝑅 → (𝑅 ⊆ (𝑆 ∩ (dom 𝑅 × ran 𝑅)) ↔ 𝑅 ⊆ 𝑆))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∩ cin 3901   ⊆ wss 3902   × cxp 5657  dom cdm 5659  ran crn 5660  Rel wrel 5664

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-cnv 5667  df-dm 5669  df-rn 5670

This theorem is used by:  cnvref4  39085