Theorem xpintrreld 44759

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 15138	. . 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 44758	1 ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ⊆ wss 3895   ∩ cin 4089   × cxp 5648   ∘ ccom 5654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662

This theorem is used by:  restrreld  44760