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Theorem cnvref4 36743
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 36740 . . . . . . 7 (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
21biimpi 215 . . . . . 6 (Rel 𝑅 → (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
32dmeqd 5860 . . . . 5 (Rel 𝑅 → dom (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = dom 𝑅)
42rneqd 5892 . . . . 5 (Rel 𝑅 → ran (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = ran 𝑅)
53, 4xpeq12d 5663 . . . 4 (Rel 𝑅 → (dom (𝑅 ∩ (dom 𝑅 × ran 𝑅)) × ran (𝑅 ∩ (dom 𝑅 × ran 𝑅))) = (dom 𝑅 × ran 𝑅))
65ineq2d 4171 . . 3 (Rel 𝑅 → (𝑆 ∩ (dom (𝑅 ∩ (dom 𝑅 × ran 𝑅)) × ran (𝑅 ∩ (dom 𝑅 × ran 𝑅)))) = (𝑆 ∩ (dom 𝑅 × ran 𝑅)))
76sseq2d 3975 . 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 5650 . . . 4 Rel (dom 𝑅 × ran 𝑅)
9 relin2 5768 . . . 4 (Rel (dom 𝑅 × ran 𝑅) → Rel (𝑅 ∩ (dom 𝑅 × ran 𝑅)))
10 relssinxpdmrn 36742 . . . 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 3974 . . 3 (Rel 𝑅 → ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑆𝑅𝑆))
1311, 12bitrid 283 . 2 (Rel 𝑅 → ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑆 ∩ (dom (𝑅 ∩ (dom 𝑅 × ran 𝑅)) × ran (𝑅 ∩ (dom 𝑅 × ran 𝑅)))) ↔ 𝑅𝑆))
147, 13bitr3d 281 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 1542  cin 3908  wss 3909   × cxp 5630  dom cdm 5632  ran crn 5633  Rel wrel 5637
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-10 2138  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
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-nf 1787  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-xp 5638  df-rel 5639  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644
This theorem is referenced by:  dfcnvrefrel4  36926
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