Theorem cossididresb 38220

Description: When 𝐵 is single-valued (see cocnv2ssid 38207) and 𝐴 is injective (see cocnv1ssid 38204) , the composition (𝐵 ∘ 𝐴) is included in the identity relation if and only if (𝐴 ∘ 𝐵) includes the identity relation restricted to (dom 𝐵 ∩ ran 𝐴). (Contributed by BJ, 8-Nov-2026.)

Assertion

Ref	Expression
cossididresb	⊢ (((𝐵 ∘ ◡𝐵) ⊆ I ∧ (◡𝐴 ∘ 𝐴) ⊆ I ) → ((𝐵 ∘ 𝐴) ⊆ I ↔ ( I ↾ (dom 𝐵 ∩ ran 𝐴)) ⊆ (𝐴 ∘ 𝐵)))

Proof of Theorem cossididresb

Step	Hyp	Ref	Expression
1		cossididres 38217	. 2 ⊢ ((𝐵 ∘ 𝐴) ⊆ I → ( I ↾ (dom 𝐵 ∩ ran 𝐴)) ⊆ (𝐴 ∘ 𝐵))
2		cnvss 5860	. . . 4 ⊢ (( I ↾ (dom 𝐵 ∩ ran 𝐴)) ⊆ (𝐴 ∘ 𝐵) → ◡( I ↾ (dom 𝐵 ∩ ran 𝐴)) ⊆ ◡(𝐴 ∘ 𝐵))
3		cnvresid 6285	. . . 4 ⊢ ◡( I ↾ (dom 𝐵 ∩ ran 𝐴)) = ( I ↾ (dom 𝐵 ∩ ran 𝐴))
4		cnvco 5877	. . . 4 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴)
5	2, 3, 4	3sstr3g 3965	. . 3 ⊢ (( I ↾ (dom 𝐵 ∩ ran 𝐴)) ⊆ (𝐴 ∘ 𝐵) → ( I ↾ (dom 𝐵 ∩ ran 𝐴)) ⊆ (◡𝐵 ∘ ◡𝐴))
6		cossidmid 38219	. . 3 ⊢ (((𝐵 ∘ ◡𝐵) ⊆ I ∧ (◡𝐴 ∘ 𝐴) ⊆ I ) → (( I ↾ (dom 𝐵 ∩ ran 𝐴)) ⊆ (◡𝐵 ∘ ◡𝐴) → (𝐵 ∘ 𝐴) ⊆ I ))
7	5, 6	syl5 35	. 2 ⊢ (((𝐵 ∘ ◡𝐵) ⊆ I ∧ (◡𝐴 ∘ 𝐴) ⊆ I ) → (( I ↾ (dom 𝐵 ∩ ran 𝐴)) ⊆ (𝐴 ∘ 𝐵) → (𝐵 ∘ 𝐴) ⊆ I ))
8	1, 7	impbid2 229	1 ⊢ (((𝐵 ∘ ◡𝐵) ⊆ I ∧ (◡𝐴 ∘ 𝐴) ⊆ I ) → ((𝐵 ∘ 𝐴) ⊆ I ↔ ( I ↾ (dom 𝐵 ∩ ran 𝐴)) ⊆ (𝐴 ∘ 𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ⊆ 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: (None)