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Theorem relssinxpdmrn 36562
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)) ↔ 𝑅𝑆))

Proof of Theorem relssinxpdmrn
StepHypRef Expression
1 relssdmrn 6186 . . 3 (Rel 𝑅𝑅 ⊆ (dom 𝑅 × ran 𝑅))
21biantrud 533 . 2 (Rel 𝑅 → (𝑅𝑆 ↔ (𝑅𝑆𝑅 ⊆ (dom 𝑅 × ran 𝑅))))
3 ssin 4170 . 2 ((𝑅𝑆𝑅 ⊆ (dom 𝑅 × ran 𝑅)) ↔ 𝑅 ⊆ (𝑆 ∩ (dom 𝑅 × ran 𝑅)))
42, 3bitr2di 288 1 (Rel 𝑅 → (𝑅 ⊆ (𝑆 ∩ (dom 𝑅 × ran 𝑅)) ↔ 𝑅𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  cin 3891  wss 3892   × cxp 5598  dom cdm 5600  ran crn 5601  Rel wrel 5605
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rex 3071  df-rab 3333  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-br 5082  df-opab 5144  df-xp 5606  df-rel 5607  df-cnv 5608  df-dm 5610  df-rn 5611
This theorem is referenced by:  cnvref4  36563
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