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Theorem trrelind 40141
Description: The intersection of transitive relations is a transitive relation. (Contributed by RP, 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 4179 . . . . 5 (𝑅𝑆) ⊆ 𝑅
32a1i 11 . . . 4 (𝜑 → (𝑅𝑆) ⊆ 𝑅)
41, 3, 3trrelssd 14309 . . 3 (𝜑 → ((𝑅𝑆) ∘ (𝑅𝑆)) ⊆ 𝑅)
5 trrelind.s . . . 4 (𝜑 → (𝑆𝑆) ⊆ 𝑆)
6 inss2 4180 . . . . 5 (𝑅𝑆) ⊆ 𝑆
76a1i 11 . . . 4 (𝜑 → (𝑅𝑆) ⊆ 𝑆)
85, 7, 7trrelssd 14309 . . 3 (𝜑 → ((𝑅𝑆) ∘ (𝑅𝑆)) ⊆ 𝑆)
94, 8ssind 4183 . 2 (𝜑 → ((𝑅𝑆) ∘ (𝑅𝑆)) ⊆ (𝑅𝑆))
10 trrelind.t . . 3 (𝜑𝑇 = (𝑅𝑆))
1110, 10coeq12d 5707 . 2 (𝜑 → (𝑇𝑇) = ((𝑅𝑆) ∘ (𝑅𝑆)))
129, 11, 103sstr4d 3989 1 (𝜑 → (𝑇𝑇) ⊆ 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  cin 3908  wss 3909  ccom 5531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-rab 3134  df-v 3472  df-in 3916  df-ss 3926  df-br 5039  df-opab 5101  df-co 5536
This theorem is referenced by:  xpintrreld  40142
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