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Theorem relssinxpdmrn 38330
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 6289 . . 3 (Rel 𝑅𝑅 ⊆ (dom 𝑅 × ran 𝑅))
21biantrud 531 . 2 (Rel 𝑅 → (𝑅𝑆 ↔ (𝑅𝑆𝑅 ⊆ (dom 𝑅 × ran 𝑅))))
3 ssin 4246 . 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 206  wa 395  cin 3961  wss 3962   × cxp 5686  dom cdm 5688  ran crn 5689  Rel wrel 5693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-xp 5694  df-rel 5695  df-cnv 5696  df-dm 5698  df-rn 5699
This theorem is referenced by:  cnvref4  38331
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