Theorem cossidres1 38213

Description: If (𝐵 ∘ 𝐴) is included in the identity relation, then the restriction of 𝐵 to the range of 𝐴 is included in the converse of 𝐴. (Contributed by BJ, 8-Nov-2026.)

Assertion

Ref	Expression
cossidres1	⊢ ((𝐵 ∘ 𝐴) ⊆ I → (𝐵 ↾ ran 𝐴) ⊆ ◡𝐴)

Proof of Theorem cossidres1

Step	Hyp	Ref	Expression
1		coires1 6286	. 2 ⊢ (𝐵 ∘ ( I ↾ ran 𝐴)) = (𝐵 ↾ ran 𝐴)
2		idresrnsscocnv2 6280	. . . 4 ⊢ ( I ↾ ran 𝐴) ⊆ (𝐴 ∘ ◡𝐴)
3		coss2 5842	. . . 4 ⊢ (( I ↾ ran 𝐴) ⊆ (𝐴 ∘ ◡𝐴) → (𝐵 ∘ ( I ↾ ran 𝐴)) ⊆ (𝐵 ∘ (𝐴 ∘ ◡𝐴)))
4	2, 3	ax-mp 5	. . 3 ⊢ (𝐵 ∘ ( I ↾ ran 𝐴)) ⊆ (𝐵 ∘ (𝐴 ∘ ◡𝐴))
5		coass 6288	. . . 4 ⊢ ((𝐵 ∘ 𝐴) ∘ ◡𝐴) = (𝐵 ∘ (𝐴 ∘ ◡𝐴))
6		coss1 5841	. . . . 5 ⊢ ((𝐵 ∘ 𝐴) ⊆ I → ((𝐵 ∘ 𝐴) ∘ ◡𝐴) ⊆ ( I ∘ ◡𝐴))
7		coi2ss 38195	. . . . 5 ⊢ ( I ∘ ◡𝐴) ⊆ ◡𝐴
8	6, 7	sstrdi 3925	. . . 4 ⊢ ((𝐵 ∘ 𝐴) ⊆ I → ((𝐵 ∘ 𝐴) ∘ ◡𝐴) ⊆ ◡𝐴)
9	5, 8	eqsstrrid 3952	. . 3 ⊢ ((𝐵 ∘ 𝐴) ⊆ I → (𝐵 ∘ (𝐴 ∘ ◡𝐴)) ⊆ ◡𝐴)
10	4, 9	sstrid 3924	. 2 ⊢ ((𝐵 ∘ 𝐴) ⊆ I → (𝐵 ∘ ( I ↾ ran 𝐴)) ⊆ ◡𝐴)
11	1, 10	eqsstrrid 3952	1 ⊢ ((𝐵 ∘ 𝐴) ⊆ I → (𝐵 ↾ ran 𝐴) ⊆ ◡𝐴)

This proof depends on syntax axioms:   → wi 4   ⊆ wss 3900   I cid 5550  ^◡ccnv 5655  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:  cossidres2  38214  cossidcnvss2  38215  cossididres  38217  nvo3  38223