Theorem xpintrreld 44254

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 15007	. . 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 44253	1 ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆)

Syntax hints:   → wi 4   = wceq 1563   ∩ cin 3906   ⊆ wss 3907   × cxp 5650   ∘ ccom 5656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-11 2194  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664

This theorem is referenced by:  restrreld  44255