Theorem coindirss 38199

Description: Precomposition is subdistributive over intersection. Compare with coundir 6266. See coindir 38210 for equality when precomposing with a single-valued class. (Contributed by BJ, 8-Nov-2026.)

Assertion

Ref	Expression
coindirss	⊢ ((𝐴 ∩ 𝐵) ∘ 𝐶) ⊆ ((𝐴 ∘ 𝐶) ∩ (𝐵 ∘ 𝐶))

Proof of Theorem coindirss

Step	Hyp	Ref	Expression
1		inss1 4136	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2		coss1 5841	. . 3 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐴 → ((𝐴 ∩ 𝐵) ∘ 𝐶) ⊆ (𝐴 ∘ 𝐶))
3	1, 2	ax-mp 5	. 2 ⊢ ((𝐴 ∩ 𝐵) ∘ 𝐶) ⊆ (𝐴 ∘ 𝐶)
4		inss2 4137	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
5		coss1 5841	. . 3 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐵 → ((𝐴 ∩ 𝐵) ∘ 𝐶) ⊆ (𝐵 ∘ 𝐶))
6	4, 5	ax-mp 5	. 2 ⊢ ((𝐴 ∩ 𝐵) ∘ 𝐶) ⊆ (𝐵 ∘ 𝐶)
7	3, 6	ssini 4139	1 ⊢ ((𝐴 ∩ 𝐵) ∘ 𝐶) ⊆ ((𝐴 ∘ 𝐶) ∩ (𝐵 ∘ 𝐶))

This proof depends on syntax axioms:   ⊆ wss 3900   ∩ cin 4092   ∘ ccom 5660

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-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-in 4094  df-br 5107  df-opab 5172  df-co 5665

This theorem is used by:  coindir  38210