Theorem coss12d 5845

Description: Subset deduction for composition of two classes. (Contributed by RP, 24-Dec-2019.) (Proof shortened by BJ, 8-Nov-2026.)

Hypotheses

Ref	Expression
coss12d.a	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
coss12d.c	⊢ (𝜑 → 𝐶 ⊆ 𝐷)

Assertion

Ref	Expression
coss12d	⊢ (𝜑 → (𝐴 ∘ 𝐶) ⊆ (𝐵 ∘ 𝐷))

Proof of Theorem coss12d

Step	Hyp	Ref	Expression
1		coss12d.a	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		coss1 5843	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∘ 𝐶) ⊆ (𝐵 ∘ 𝐶))
3	1, 2	syl 18	. 2 ⊢ (𝜑 → (𝐴 ∘ 𝐶) ⊆ (𝐵 ∘ 𝐶))
4		coss12d.c	. . 3 ⊢ (𝜑 → 𝐶 ⊆ 𝐷)
5		coss2 5844	. . 3 ⊢ (𝐶 ⊆ 𝐷 → (𝐵 ∘ 𝐶) ⊆ (𝐵 ∘ 𝐷))
6	4, 5	syl 18	. 2 ⊢ (𝜑 → (𝐵 ∘ 𝐶) ⊆ (𝐵 ∘ 𝐷))
7	3, 6	sstrd 3923	1 ⊢ (𝜑 → (𝐴 ∘ 𝐶) ⊆ (𝐵 ∘ 𝐷))

This proof depends on syntax axioms:   → wi 4   ⊆ wss 3900   ∘ ccom 5662

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ss 3901  df-br 5109  df-opab 5174  df-co 5667

This theorem is used by:  trrelssd  15169  ustund  24596  coindir  38217  coindi  38219  cossidcnvss2  38222  cossididres  38224  cossidmid  38226  nvo3  38230  bj-imdirco  38379  relexpss1d  44966