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Theorem trrelind 38455
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 3972 . . . . 5 (𝑅𝑆) ⊆ 𝑅
32a1i 11 . . . 4 (𝜑 → (𝑅𝑆) ⊆ 𝑅)
41, 3, 3trrelssd 13909 . . 3 (𝜑 → ((𝑅𝑆) ∘ (𝑅𝑆)) ⊆ 𝑅)
5 trrelind.s . . . 4 (𝜑 → (𝑆𝑆) ⊆ 𝑆)
6 inss2 3973 . . . . 5 (𝑅𝑆) ⊆ 𝑆
76a1i 11 . . . 4 (𝜑 → (𝑅𝑆) ⊆ 𝑆)
85, 7, 7trrelssd 13909 . . 3 (𝜑 → ((𝑅𝑆) ∘ (𝑅𝑆)) ⊆ 𝑆)
94, 8ssind 3976 . 2 (𝜑 → ((𝑅𝑆) ∘ (𝑅𝑆)) ⊆ (𝑅𝑆))
10 trrelind.t . . 3 (𝜑𝑇 = (𝑅𝑆))
1110, 10coeq12d 5438 . 2 (𝜑 → (𝑇𝑇) = ((𝑅𝑆) ∘ (𝑅𝑆)))
129, 11, 103sstr4d 3785 1 (𝜑 → (𝑇𝑇) ⊆ 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1628  cin 3710  wss 3711  ccom 5266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-v 3338  df-in 3718  df-ss 3725  df-br 4801  df-opab 4861  df-co 5271
This theorem is referenced by:  xpintrreld  38456
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