Theorem cossidmid 38219

Description: If (𝐴 ∘ 𝐵) and (𝐶 ∘ 𝐷) are included in the identity relation and (𝐵 ∘ 𝐶) includes it on (dom 𝐴 ∩ ran 𝐷), then (𝐴 ∘ 𝐷) is included in the identity relation. (Contributed by BJ, 8-Nov-2026.)

Assertion

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

Proof of Theorem cossidmid

Step	Hyp	Ref	Expression
1		coidmid 38218	. . 3 ⊢ (( I ↾ (dom 𝐴 ∩ ran 𝐷)) ⊆ (𝐵 ∘ 𝐶) → (𝐴 ∘ 𝐷) ⊆ ((𝐴 ∘ 𝐵) ∘ (𝐶 ∘ 𝐷)))
2		simpl 488	. . . . 5 ⊢ (((𝐴 ∘ 𝐵) ⊆ I ∧ (𝐶 ∘ 𝐷) ⊆ I ) → (𝐴 ∘ 𝐵) ⊆ I )
3		simpr 490	. . . . 5 ⊢ (((𝐴 ∘ 𝐵) ⊆ I ∧ (𝐶 ∘ 𝐷) ⊆ I ) → (𝐶 ∘ 𝐷) ⊆ I )
4	2, 3	coss12d 5843	. . . 4 ⊢ (((𝐴 ∘ 𝐵) ⊆ I ∧ (𝐶 ∘ 𝐷) ⊆ I ) → ((𝐴 ∘ 𝐵) ∘ (𝐶 ∘ 𝐷)) ⊆ ( I ∘ I ))
5		reli 5812	. . . . 5 ⊢ Rel I
6		coi1 6283	. . . . 5 ⊢ (Rel I → ( I ∘ I ) = I )
7	5, 6	ax-mp 5	. . . 4 ⊢ ( I ∘ I ) = I
8	4, 7	sseqtrdi 3953	. . 3 ⊢ (((𝐴 ∘ 𝐵) ⊆ I ∧ (𝐶 ∘ 𝐷) ⊆ I ) → ((𝐴 ∘ 𝐵) ∘ (𝐶 ∘ 𝐷)) ⊆ I )
9	1, 8	sylan9ssr 3927	. 2 ⊢ ((((𝐴 ∘ 𝐵) ⊆ I ∧ (𝐶 ∘ 𝐷) ⊆ I ) ∧ ( I ↾ (dom 𝐴 ∩ ran 𝐷)) ⊆ (𝐵 ∘ 𝐶)) → (𝐴 ∘ 𝐷) ⊆ I )
10	9	ex 418	1 ⊢ (((𝐴 ∘ 𝐵) ⊆ I ∧ (𝐶 ∘ 𝐷) ⊆ I ) → (( I ↾ (dom 𝐴 ∩ ran 𝐷)) ⊆ (𝐵 ∘ 𝐶) → (𝐴 ∘ 𝐷) ⊆ I ))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ⊆ wss 3900   ∩ cin 4092   I cid 5550  dom cdm 5656  ran crn 5657   ↾ cres 5658   ∘ ccom 5660  Rel wrel 5661

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