Theorem cossididres 38217

Description: If a composition is included in the identity relation, then the reverse composition includes the identity relation restricted to the appropriate class. (Contributed by BJ, 7-Nov-2026.)

Assertion

Ref	Expression
cossididres	⊢ ((𝐵 ∘ 𝐴) ⊆ I → ( I ↾ (dom 𝐵 ∩ ran 𝐴)) ⊆ (𝐴 ∘ 𝐵))

Proof of Theorem cossididres

Step	Hyp	Ref	Expression
1		dmres 6018	. . . . 5 ⊢ dom (𝐵 ↾ ran 𝐴) = (ran 𝐴 ∩ dom 𝐵)
2		incom 4109	. . . . 5 ⊢ (ran 𝐴 ∩ dom 𝐵) = (dom 𝐵 ∩ ran 𝐴)
3	1, 2	eqtri 2786	. . . 4 ⊢ dom (𝐵 ↾ ran 𝐴) = (dom 𝐵 ∩ ran 𝐴)
4	3	reseq2i 5981	. . 3 ⊢ ( I ↾ dom (𝐵 ↾ ran 𝐴)) = ( I ↾ (dom 𝐵 ∩ ran 𝐴))
5		idresdmsscocnv1 6279	. . 3 ⊢ ( I ↾ dom (𝐵 ↾ ran 𝐴)) ⊆ (◡(𝐵 ↾ ran 𝐴) ∘ (𝐵 ↾ ran 𝐴))
6	4, 5	eqsstrri 3960	. 2 ⊢ ( I ↾ (dom 𝐵 ∩ ran 𝐴)) ⊆ (◡(𝐵 ↾ ran 𝐴) ∘ (𝐵 ↾ ran 𝐴))
7		cossidres1 38213	. . . . 5 ⊢ ((𝐵 ∘ 𝐴) ⊆ I → (𝐵 ↾ ran 𝐴) ⊆ ◡𝐴)
8		cnvss 5860	. . . . 5 ⊢ ((𝐵 ↾ ran 𝐴) ⊆ ◡𝐴 → ◡(𝐵 ↾ ran 𝐴) ⊆ ◡◡𝐴)
9	7, 8	syl 18	. . . 4 ⊢ ((𝐵 ∘ 𝐴) ⊆ I → ◡(𝐵 ↾ ran 𝐴) ⊆ ◡◡𝐴)
10		resss 6006	. . . . 5 ⊢ (𝐵 ↾ ran 𝐴) ⊆ 𝐵
11	10	a1i 11	. . . 4 ⊢ ((𝐵 ∘ 𝐴) ⊆ I → (𝐵 ↾ ran 𝐴) ⊆ 𝐵)
12	9, 11	coss12d 5843	. . 3 ⊢ ((𝐵 ∘ 𝐴) ⊆ I → (◡(𝐵 ↾ ran 𝐴) ∘ (𝐵 ↾ ran 𝐴)) ⊆ (◡◡𝐴 ∘ 𝐵))
13		cocnvcnv1 6276	. . 3 ⊢ (◡◡𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵)
14	12, 13	sseqtrdi 3953	. 2 ⊢ ((𝐵 ∘ 𝐴) ⊆ I → (◡(𝐵 ↾ ran 𝐴) ∘ (𝐵 ↾ ran 𝐴)) ⊆ (𝐴 ∘ 𝐵))
15	6, 14	sstrid 3924	1 ⊢ ((𝐵 ∘ 𝐴) ⊆ I → ( I ↾ (dom 𝐵 ∩ ran 𝐴)) ⊆ (𝐴 ∘ 𝐵))

This proof depends on syntax axioms:   → wi 4   ⊆ wss 3900   ∩ cin 4092   I cid 5550  ^◡ccnv 5655  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:  cossididresb  38220  nvo  38221