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Theorem relssinxpdmrn 37156
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 6264 . . 3 (Rel 𝑅𝑅 ⊆ (dom 𝑅 × ran 𝑅))
21biantrud 533 . 2 (Rel 𝑅 → (𝑅𝑆 ↔ (𝑅𝑆𝑅 ⊆ (dom 𝑅 × ran 𝑅))))
3 ssin 4229 . 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 3946  wss 3947   × cxp 5673  dom cdm 5675  ran crn 5676  Rel wrel 5680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-cnv 5683  df-dm 5685  df-rn 5686
This theorem is referenced by:  cnvref4  37157
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