Theorem restrreld 44023

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 5644	. . 3 ⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × V))
4	2, 3	eqtrdi 2788	. 2 ⊢ (𝜑 → 𝑆 = (𝑅 ∩ (𝐴 × V)))
5	1, 4	xpintrreld 44022	1 ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆)

Syntax hints:   → wi 4   = wceq 1542  Vcvv 3442   ∩ cin 3902   ⊆ wss 3903   × cxp 5630   ↾ cres 5634   ∘ ccom 5636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644

This theorem is referenced by: (None)