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Theorem trrelind 36873
Description: The intersection of transitive relations is a transitive relation. (Contributed by Richard Penner, 24-Dec-2019.)
Hypotheses
Ref Expression
trrelind.r (𝜑 → (𝑅𝑅) ⊆ 𝑅)
trrelind.s (𝜑 → (𝑆𝑆) ⊆ 𝑆)
trrelind.t (𝜑𝑇 = (𝑅𝑆))
Assertion
Ref Expression
trrelind (𝜑 → (𝑇𝑇) ⊆ 𝑇)

Proof of Theorem trrelind
StepHypRef Expression
1 trrelind.r . . . 4 (𝜑 → (𝑅𝑅) ⊆ 𝑅)
2 inss1 3698 . . . . 5 (𝑅𝑆) ⊆ 𝑅
32a1i 11 . . . 4 (𝜑 → (𝑅𝑆) ⊆ 𝑅)
41, 3, 3trrelssd 13419 . . 3 (𝜑 → ((𝑅𝑆) ∘ (𝑅𝑆)) ⊆ 𝑅)
5 trrelind.s . . . 4 (𝜑 → (𝑆𝑆) ⊆ 𝑆)
6 inss2 3699 . . . . 5 (𝑅𝑆) ⊆ 𝑆
76a1i 11 . . . 4 (𝜑 → (𝑅𝑆) ⊆ 𝑆)
85, 7, 7trrelssd 13419 . . 3 (𝜑 → ((𝑅𝑆) ∘ (𝑅𝑆)) ⊆ 𝑆)
94, 8ssind 3702 . 2 (𝜑 → ((𝑅𝑆) ∘ (𝑅𝑆)) ⊆ (𝑅𝑆))
10 trrelind.t . . 3 (𝜑𝑇 = (𝑅𝑆))
1110, 10coeq12d 5100 . 2 (𝜑 → (𝑇𝑇) = ((𝑅𝑆) ∘ (𝑅𝑆)))
129, 11, 103sstr4d 3515 1 (𝜑 → (𝑇𝑇) ⊆ 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  cin 3443  wss 3444  ccom 4936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-v 3079  df-in 3451  df-ss 3458  df-br 4482  df-opab 4542  df-co 4941
This theorem is referenced by:  xpintrreld  36874
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