Theorem coidmid 38218

Description: A composition that includes the identity relation on the relevant class can be inserted in the middle of a composition. (Contributed by BJ, 8-Nov-2026.)

Assertion

Ref	Expression
coidmid	⊢ (( I ↾ (dom 𝐴 ∩ ran 𝐷)) ⊆ (𝐵 ∘ 𝐶) → (𝐴 ∘ 𝐷) ⊆ ((𝐴 ∘ 𝐵) ∘ (𝐶 ∘ 𝐷)))

Proof of Theorem coidmid

Step	Hyp	Ref	Expression
1		coss2 5842	. . 3 ⊢ (( I ↾ (dom 𝐴 ∩ ran 𝐷)) ⊆ (𝐵 ∘ 𝐶) → (𝐴 ∘ ( I ↾ (dom 𝐴 ∩ ran 𝐷))) ⊆ (𝐴 ∘ (𝐵 ∘ 𝐶)))
2		coss1 5841	. . 3 ⊢ ((𝐴 ∘ ( I ↾ (dom 𝐴 ∩ ran 𝐷))) ⊆ (𝐴 ∘ (𝐵 ∘ 𝐶)) → ((𝐴 ∘ ( I ↾ (dom 𝐴 ∩ ran 𝐷))) ∘ 𝐷) ⊆ ((𝐴 ∘ (𝐵 ∘ 𝐶)) ∘ 𝐷))
3	1, 2	syl 18	. 2 ⊢ (( I ↾ (dom 𝐴 ∩ ran 𝐷)) ⊆ (𝐵 ∘ 𝐶) → ((𝐴 ∘ ( I ↾ (dom 𝐴 ∩ ran 𝐷))) ∘ 𝐷) ⊆ ((𝐴 ∘ (𝐵 ∘ 𝐶)) ∘ 𝐷))
4		coires1 6286	. . . . 5 ⊢ (𝐴 ∘ ( I ↾ (dom 𝐴 ∩ ran 𝐷))) = (𝐴 ↾ (dom 𝐴 ∩ ran 𝐷))
5		incom 4109	. . . . . . 7 ⊢ (dom 𝐴 ∩ ran 𝐷) = (ran 𝐷 ∩ dom 𝐴)
6	5	reseq2i 5981	. . . . . 6 ⊢ (𝐴 ↾ (dom 𝐴 ∩ ran 𝐷)) = (𝐴 ↾ (ran 𝐷 ∩ dom 𝐴))
7		resindm 6035	. . . . . 6 ⊢ (𝐴 ↾ (ran 𝐷 ∩ dom 𝐴)) = (𝐴 ↾ ran 𝐷)
8	6, 7	eqtri 2786	. . . . 5 ⊢ (𝐴 ↾ (dom 𝐴 ∩ ran 𝐷)) = (𝐴 ↾ ran 𝐷)
9	4, 8	eqtri 2786	. . . 4 ⊢ (𝐴 ∘ ( I ↾ (dom 𝐴 ∩ ran 𝐷))) = (𝐴 ↾ ran 𝐷)
10	9	coeq1i 5846	. . 3 ⊢ ((𝐴 ∘ ( I ↾ (dom 𝐴 ∩ ran 𝐷))) ∘ 𝐷) = ((𝐴 ↾ ran 𝐷) ∘ 𝐷)
11		coresrn 38197	. . 3 ⊢ ((𝐴 ↾ ran 𝐷) ∘ 𝐷) = (𝐴 ∘ 𝐷)
12	10, 11	eqtri 2786	. 2 ⊢ ((𝐴 ∘ ( I ↾ (dom 𝐴 ∩ ran 𝐷))) ∘ 𝐷) = (𝐴 ∘ 𝐷)
13		coass 6288	. . . 4 ⊢ ((𝐴 ∘ 𝐵) ∘ 𝐶) = (𝐴 ∘ (𝐵 ∘ 𝐶))
14	13	coeq1i 5846	. . 3 ⊢ (((𝐴 ∘ 𝐵) ∘ 𝐶) ∘ 𝐷) = ((𝐴 ∘ (𝐵 ∘ 𝐶)) ∘ 𝐷)
15		coass 6288	. . 3 ⊢ (((𝐴 ∘ 𝐵) ∘ 𝐶) ∘ 𝐷) = ((𝐴 ∘ 𝐵) ∘ (𝐶 ∘ 𝐷))
16	14, 15	eqtr3i 2788	. 2 ⊢ ((𝐴 ∘ (𝐵 ∘ 𝐶)) ∘ 𝐷) = ((𝐴 ∘ 𝐵) ∘ (𝐶 ∘ 𝐷))
17	3, 12, 16	3sstr3g 3965	1 ⊢ (( I ↾ (dom 𝐴 ∩ ran 𝐷)) ⊆ (𝐵 ∘ 𝐶) → (𝐴 ∘ 𝐷) ⊆ ((𝐴 ∘ 𝐵) ∘ (𝐶 ∘ 𝐷)))

This proof depends on syntax axioms:   → wi 4   ⊆ wss 3900   ∩ cin 4092   I cid 5550  dom cdm 5656  ran crn 5657   ↾ cres 5658   ∘ 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-dm 5666  df-rn 5667  df-res 5668

This theorem is used by:  cossidmid  38219