Theorem restrreld 44243

Description: The restriction of a transitive relation is a transitive relation. (Contributed by RP, 24-Dec-2019.)

Hypotheses

Ref	Expression
restrreld.r	⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅)
restrreld.s	⊢ (𝜑 → 𝑆 = (𝑅 ↾ 𝐴))

Assertion

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

Proof of Theorem restrreld

Step	Hyp	Ref	Expression
1		restrreld.r	. 2 ⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅)
2		restrreld.s	. . 3 ⊢ (𝜑 → 𝑆 = (𝑅 ↾ 𝐴))
3		df-res 5659	. . 3 ⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × V))
4	2, 3	eqtrdi 2813	. 2 ⊢ (𝜑 → 𝑆 = (𝑅 ∩ (𝐴 × V)))
5	1, 4	xpintrreld 44242	1 ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆)

Syntax hints:   → wi 4   = wceq 1560  Vcvv 3454   ∩ cin 3903   ⊆ wss 3904   × cxp 5645   ↾ cres 5649   ∘ ccom 5651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-11 2191  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659

This theorem is referenced by: (None)