Theorem xpintrreld 42998

Description: The intersection of a transitive relation with a Cartesian product is a transitive relation. (Contributed by RP, 24-Dec-2019.)

Hypotheses

Ref	Expression
xpintrreld.r	⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅)
xpintrreld.s	⊢ (𝜑 → 𝑆 = (𝑅 ∩ (𝐴 × 𝐵)))

Assertion

Ref	Expression
xpintrreld	⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆)

Proof of Theorem xpintrreld

Step	Hyp	Ref	Expression
1		xpintrreld.r	. 2 ⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅)
2		xptrrel 14933	. . 3 ⊢ ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)
3	2	a1i 11	. 2 ⊢ (𝜑 → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵))
4		xpintrreld.s	. 2 ⊢ (𝜑 → 𝑆 = (𝑅 ∩ (𝐴 × 𝐵)))
5	1, 3, 4	trrelind 42997	1 ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆)

Syntax hints:   → wi 4   = wceq 1533   ∩ cin 3942   ⊆ wss 3943   × cxp 5667   ∘ ccom 5673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681

This theorem is referenced by:  restrreld  42999