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Theorem relssinxpdmrn 37708
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 6257 . . 3 (Rel 𝑅𝑅 ⊆ (dom 𝑅 × ran 𝑅))
21biantrud 531 . 2 (Rel 𝑅 → (𝑅𝑆 ↔ (𝑅𝑆𝑅 ⊆ (dom 𝑅 × ran 𝑅))))
3 ssin 4222 . 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 395  cin 3939  wss 3940   × cxp 5664  dom cdm 5666  ran crn 5667  Rel wrel 5671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-xp 5672  df-rel 5673  df-cnv 5674  df-dm 5676  df-rn 5677
This theorem is referenced by:  cnvref4  37709
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