Theorem restrreld 36978

Description: The restriction of a transitive relation is a transitive relation. (Contributed by Richard Penner, 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 5050	. . 3 ⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × V))
4	2, 3	syl6eq 2660	. 2 ⊢ (𝜑 → 𝑆 = (𝑅 ∩ (𝐴 × V)))
5	1, 4	xpintrreld 36977	1 ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆)

Syntax hints:   → wi 4   = wceq 1475  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540   × cxp 5036   ↾ cres 5040   ∘ ccom 5042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050

This theorem is referenced by: (None)