Theorem cossidres2 38214

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

Assertion

Ref	Expression
cossidres2	⊢ ((𝐵 ∘ 𝐴) ⊆ I → (◡𝐴 ↾ dom 𝐵) ⊆ 𝐵)

Proof of Theorem cossidres2

Step	Hyp	Ref	Expression
1		dfdm4 5887	. . 3 ⊢ dom 𝐵 = ran ◡𝐵
2	1	reseq2i 5981	. 2 ⊢ (◡𝐴 ↾ dom 𝐵) = (◡𝐴 ↾ ran ◡𝐵)
3		cnvco 5877	. . . . 5 ⊢ ◡(𝐵 ∘ 𝐴) = (◡𝐴 ∘ ◡𝐵)
4		cnvss 5860	. . . . . 6 ⊢ ((𝐵 ∘ 𝐴) ⊆ I → ◡(𝐵 ∘ 𝐴) ⊆ ◡ I )
5		cnvi 5873	. . . . . 6 ⊢ ◡ I = I
6	4, 5	sseqtrdi 3953	. . . . 5 ⊢ ((𝐵 ∘ 𝐴) ⊆ I → ◡(𝐵 ∘ 𝐴) ⊆ I )
7	3, 6	eqsstrrid 3952	. . . 4 ⊢ ((𝐵 ∘ 𝐴) ⊆ I → (◡𝐴 ∘ ◡𝐵) ⊆ I )
8		cossidres1 38213	. . . 4 ⊢ ((◡𝐴 ∘ ◡𝐵) ⊆ I → (◡𝐴 ↾ ran ◡𝐵) ⊆ ◡◡𝐵)
9	7, 8	syl 18	. . 3 ⊢ ((𝐵 ∘ 𝐴) ⊆ I → (◡𝐴 ↾ ran ◡𝐵) ⊆ ◡◡𝐵)
10		cnvcnvss 6202	. . 3 ⊢ ◡◡𝐵 ⊆ 𝐵
11	9, 10	sstrdi 3925	. 2 ⊢ ((𝐵 ∘ 𝐴) ⊆ I → (◡𝐴 ↾ ran ◡𝐵) ⊆ 𝐵)
12	2, 11	eqsstrid 3951	1 ⊢ ((𝐵 ∘ 𝐴) ⊆ I → (◡𝐴 ↾ dom 𝐵) ⊆ 𝐵)

This proof depends on syntax axioms:   → wi 4   ⊆ wss 3900   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:  cossidcnvss2  38215  cossidcnvss  38216  nvo3  38223