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Theorem cnvref4 37305
Description: Two ways to say that a relation is a subclass. (Contributed by Peter Mazsa, 11-Apr-2023.)
Assertion
Ref Expression
cnvref4 (Rel 𝑅 → ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑆 ∩ (dom 𝑅 × ran 𝑅)) ↔ 𝑅𝑆))

Proof of Theorem cnvref4
StepHypRef Expression
1 dfrel6 37302 . . . . . . 7 (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
21biimpi 215 . . . . . 6 (Rel 𝑅 → (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
32dmeqd 5905 . . . . 5 (Rel 𝑅 → dom (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = dom 𝑅)
42rneqd 5937 . . . . 5 (Rel 𝑅 → ran (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = ran 𝑅)
53, 4xpeq12d 5707 . . . 4 (Rel 𝑅 → (dom (𝑅 ∩ (dom 𝑅 × ran 𝑅)) × ran (𝑅 ∩ (dom 𝑅 × ran 𝑅))) = (dom 𝑅 × ran 𝑅))
65ineq2d 4212 . . 3 (Rel 𝑅 → (𝑆 ∩ (dom (𝑅 ∩ (dom 𝑅 × ran 𝑅)) × ran (𝑅 ∩ (dom 𝑅 × ran 𝑅)))) = (𝑆 ∩ (dom 𝑅 × ran 𝑅)))
76sseq2d 4014 . 2 (Rel 𝑅 → ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑆 ∩ (dom (𝑅 ∩ (dom 𝑅 × ran 𝑅)) × ran (𝑅 ∩ (dom 𝑅 × ran 𝑅)))) ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑆 ∩ (dom 𝑅 × ran 𝑅))))
8 relxp 5694 . . . 4 Rel (dom 𝑅 × ran 𝑅)
9 relin2 5813 . . . 4 (Rel (dom 𝑅 × ran 𝑅) → Rel (𝑅 ∩ (dom 𝑅 × ran 𝑅)))
10 relssinxpdmrn 37304 . . . 4 (Rel (𝑅 ∩ (dom 𝑅 × ran 𝑅)) → ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑆 ∩ (dom (𝑅 ∩ (dom 𝑅 × ran 𝑅)) × ran (𝑅 ∩ (dom 𝑅 × ran 𝑅)))) ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑆))
118, 9, 10mp2b 10 . . 3 ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑆 ∩ (dom (𝑅 ∩ (dom 𝑅 × ran 𝑅)) × ran (𝑅 ∩ (dom 𝑅 × ran 𝑅)))) ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑆)
122sseq1d 4013 . . 3 (Rel 𝑅 → ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑆𝑅𝑆))
1311, 12bitrid 282 . 2 (Rel 𝑅 → ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑆 ∩ (dom (𝑅 ∩ (dom 𝑅 × ran 𝑅)) × ran (𝑅 ∩ (dom 𝑅 × ran 𝑅)))) ↔ 𝑅𝑆))
147, 13bitr3d 280 1 (Rel 𝑅 → ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑆 ∩ (dom 𝑅 × ran 𝑅)) ↔ 𝑅𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  cin 3947  wss 3948   × cxp 5674  dom cdm 5676  ran crn 5677  Rel wrel 5681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688
This theorem is referenced by:  dfcnvrefrel4  37488
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