Theorem restrreld 44510

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 5667	. . 3 ⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × V))
4	2, 3	eqtrdi 2811	. 2 ⊢ (𝜑 → 𝑆 = (𝑅 ∩ (𝐴 × V)))
5	1, 4	xpintrreld 44509	1 ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  Vcvv 3450   ∩ cin 3898   ⊆ wss 3899   × cxp 5653   ↾ cres 5657   ∘ ccom 5659

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  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-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667

This theorem is used by: (None)