Theorem coindi 38212

Description: Postcomposition with an injective class (see cocnv1ssid 38204) distributes over intersection. (Contributed by BJ, 8-Nov-2026.)

Assertion

Ref	Expression
coindi	⊢ ((◡𝐹 ∘ 𝐹) ⊆ I → (𝐹 ∘ (𝐴 ∩ 𝐵)) = ((𝐹 ∘ 𝐴) ∩ (𝐹 ∘ 𝐵)))

Proof of Theorem coindi

Step	Hyp	Ref	Expression
1		coindiss 38198	. . 3 ⊢ (𝐹 ∘ (𝐴 ∩ 𝐵)) ⊆ ((𝐹 ∘ 𝐴) ∩ (𝐹 ∘ 𝐵))
2	1	a1i 11	. 2 ⊢ ((◡𝐹 ∘ 𝐹) ⊆ I → (𝐹 ∘ (𝐴 ∩ 𝐵)) ⊆ ((𝐹 ∘ 𝐴) ∩ (𝐹 ∘ 𝐵)))
3		comodular 38200	. . 3 ⊢ ((𝐹 ∘ 𝐴) ∩ (𝐹 ∘ 𝐵)) ⊆ ((𝐹 ∩ ((𝐹 ∘ 𝐵) ∘ ◡𝐴)) ∘ (𝐴 ∩ (◡𝐹 ∘ (𝐹 ∘ 𝐵))))
4		inss1 4136	. . . . 5 ⊢ (𝐹 ∩ ((𝐹 ∘ 𝐵) ∘ ◡𝐴)) ⊆ 𝐹
5	4	a1i 11	. . . 4 ⊢ ((◡𝐹 ∘ 𝐹) ⊆ I → (𝐹 ∩ ((𝐹 ∘ 𝐵) ∘ ◡𝐴)) ⊆ 𝐹)
6		coass 6288	. . . . . 6 ⊢ ((◡𝐹 ∘ 𝐹) ∘ 𝐵) = (◡𝐹 ∘ (𝐹 ∘ 𝐵))
7		coss1 5841	. . . . . . 7 ⊢ ((◡𝐹 ∘ 𝐹) ⊆ I → ((◡𝐹 ∘ 𝐹) ∘ 𝐵) ⊆ ( I ∘ 𝐵))
8		coi2ss 38195	. . . . . . 7 ⊢ ( I ∘ 𝐵) ⊆ 𝐵
9	7, 8	sstrdi 3925	. . . . . 6 ⊢ ((◡𝐹 ∘ 𝐹) ⊆ I → ((◡𝐹 ∘ 𝐹) ∘ 𝐵) ⊆ 𝐵)
10	6, 9	eqsstrrid 3952	. . . . 5 ⊢ ((◡𝐹 ∘ 𝐹) ⊆ I → (◡𝐹 ∘ (𝐹 ∘ 𝐵)) ⊆ 𝐵)
11		sslin 4142	. . . . 5 ⊢ ((◡𝐹 ∘ (𝐹 ∘ 𝐵)) ⊆ 𝐵 → (𝐴 ∩ (◡𝐹 ∘ (𝐹 ∘ 𝐵))) ⊆ (𝐴 ∩ 𝐵))
12	10, 11	syl 18	. . . 4 ⊢ ((◡𝐹 ∘ 𝐹) ⊆ I → (𝐴 ∩ (◡𝐹 ∘ (𝐹 ∘ 𝐵))) ⊆ (𝐴 ∩ 𝐵))
13	5, 12	coss12d 5843	. . 3 ⊢ ((◡𝐹 ∘ 𝐹) ⊆ I → ((𝐹 ∩ ((𝐹 ∘ 𝐵) ∘ ◡𝐴)) ∘ (𝐴 ∩ (◡𝐹 ∘ (𝐹 ∘ 𝐵)))) ⊆ (𝐹 ∘ (𝐴 ∩ 𝐵)))
14	3, 13	sstrid 3924	. 2 ⊢ ((◡𝐹 ∘ 𝐹) ⊆ I → ((𝐹 ∘ 𝐴) ∩ (𝐹 ∘ 𝐵)) ⊆ (𝐹 ∘ (𝐴 ∩ 𝐵)))
15	2, 14	eqssd 3930	1 ⊢ ((◡𝐹 ∘ 𝐹) ⊆ I → (𝐹 ∘ (𝐴 ∩ 𝐵)) = ((𝐹 ∘ 𝐴) ∩ (𝐹 ∘ 𝐵)))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ⊆ wss 3900   ∩ cin 4092   I cid 5550  ^◡ccnv 5655   ∘ 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  ax-sep 5253  ax-pr 5396

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-res 5668

This theorem is used by: (None)