Theorem cossidcnvss 38216

Description: A class composed with itself is included in the identity relation and has its range included in its domain if and only if its converse is included in it and it is single-valued. (Contributed by BJ, 8-Nov-2026.)

Assertion

Ref	Expression
cossidcnvss	⊢ (((𝐵 ∘ 𝐵) ⊆ I ∧ ran 𝐵 ⊆ dom 𝐵) ↔ (◡𝐵 ⊆ 𝐵 ∧ (𝐵 ∘ ◡𝐵) ⊆ I ))

Proof of Theorem cossidcnvss

Step	Hyp	Ref	Expression
1		bj-cnvresrn 38190	. . . 4 ⊢ (◡𝐵 ↾ ran 𝐵) = ◡𝐵
2		simpr 490	. . . . . 6 ⊢ (((𝐵 ∘ 𝐵) ⊆ I ∧ ran 𝐵 ⊆ dom 𝐵) → ran 𝐵 ⊆ dom 𝐵)
3		ssres2 6009	. . . . . 6 ⊢ (ran 𝐵 ⊆ dom 𝐵 → (◡𝐵 ↾ ran 𝐵) ⊆ (◡𝐵 ↾ dom 𝐵))
4	2, 3	syl 18	. . . . 5 ⊢ (((𝐵 ∘ 𝐵) ⊆ I ∧ ran 𝐵 ⊆ dom 𝐵) → (◡𝐵 ↾ ran 𝐵) ⊆ (◡𝐵 ↾ dom 𝐵))
5		simpl 488	. . . . . 6 ⊢ (((𝐵 ∘ 𝐵) ⊆ I ∧ ran 𝐵 ⊆ dom 𝐵) → (𝐵 ∘ 𝐵) ⊆ I )
6		cossidres2 38214	. . . . . 6 ⊢ ((𝐵 ∘ 𝐵) ⊆ I → (◡𝐵 ↾ dom 𝐵) ⊆ 𝐵)
7	5, 6	syl 18	. . . . 5 ⊢ (((𝐵 ∘ 𝐵) ⊆ I ∧ ran 𝐵 ⊆ dom 𝐵) → (◡𝐵 ↾ dom 𝐵) ⊆ 𝐵)
8	4, 7	sstrd 3923	. . . 4 ⊢ (((𝐵 ∘ 𝐵) ⊆ I ∧ ran 𝐵 ⊆ dom 𝐵) → (◡𝐵 ↾ ran 𝐵) ⊆ 𝐵)
9	1, 8	eqsstrrid 3952	. . 3 ⊢ (((𝐵 ∘ 𝐵) ⊆ I ∧ ran 𝐵 ⊆ dom 𝐵) → ◡𝐵 ⊆ 𝐵)
10		coss2 5842	. . . . 5 ⊢ (◡𝐵 ⊆ 𝐵 → (𝐵 ∘ ◡𝐵) ⊆ (𝐵 ∘ 𝐵))
11	9, 10	syl 18	. . . 4 ⊢ (((𝐵 ∘ 𝐵) ⊆ I ∧ ran 𝐵 ⊆ dom 𝐵) → (𝐵 ∘ ◡𝐵) ⊆ (𝐵 ∘ 𝐵))
12	11, 5	sstrd 3923	. . 3 ⊢ (((𝐵 ∘ 𝐵) ⊆ I ∧ ran 𝐵 ⊆ dom 𝐵) → (𝐵 ∘ ◡𝐵) ⊆ I )
13	9, 12	jca 521	. 2 ⊢ (((𝐵 ∘ 𝐵) ⊆ I ∧ ran 𝐵 ⊆ dom 𝐵) → (◡𝐵 ⊆ 𝐵 ∧ (𝐵 ∘ ◡𝐵) ⊆ I ))
14		cocnvcnv2 6277	. . . 4 ⊢ (𝐵 ∘ ◡◡𝐵) = (𝐵 ∘ 𝐵)
15		simpl 488	. . . . . . 7 ⊢ ((◡𝐵 ⊆ 𝐵 ∧ (𝐵 ∘ ◡𝐵) ⊆ I ) → ◡𝐵 ⊆ 𝐵)
16		cnvss 5860	. . . . . . 7 ⊢ (◡𝐵 ⊆ 𝐵 → ◡◡𝐵 ⊆ ◡𝐵)
17	15, 16	syl 18	. . . . . 6 ⊢ ((◡𝐵 ⊆ 𝐵 ∧ (𝐵 ∘ ◡𝐵) ⊆ I ) → ◡◡𝐵 ⊆ ◡𝐵)
18		coss2 5842	. . . . . 6 ⊢ (◡◡𝐵 ⊆ ◡𝐵 → (𝐵 ∘ ◡◡𝐵) ⊆ (𝐵 ∘ ◡𝐵))
19	17, 18	syl 18	. . . . 5 ⊢ ((◡𝐵 ⊆ 𝐵 ∧ (𝐵 ∘ ◡𝐵) ⊆ I ) → (𝐵 ∘ ◡◡𝐵) ⊆ (𝐵 ∘ ◡𝐵))
20		simpr 490	. . . . 5 ⊢ ((◡𝐵 ⊆ 𝐵 ∧ (𝐵 ∘ ◡𝐵) ⊆ I ) → (𝐵 ∘ ◡𝐵) ⊆ I )
21	19, 20	sstrd 3923	. . . 4 ⊢ ((◡𝐵 ⊆ 𝐵 ∧ (𝐵 ∘ ◡𝐵) ⊆ I ) → (𝐵 ∘ ◡◡𝐵) ⊆ I )
22	14, 21	eqsstrrid 3952	. . 3 ⊢ ((◡𝐵 ⊆ 𝐵 ∧ (𝐵 ∘ ◡𝐵) ⊆ I ) → (𝐵 ∘ 𝐵) ⊆ I )
23		df-rn 5667	. . . 4 ⊢ ran 𝐵 = dom ◡𝐵
24		dmss 5894	. . . . 5 ⊢ (◡𝐵 ⊆ 𝐵 → dom ◡𝐵 ⊆ dom 𝐵)
25	15, 24	syl 18	. . . 4 ⊢ ((◡𝐵 ⊆ 𝐵 ∧ (𝐵 ∘ ◡𝐵) ⊆ I ) → dom ◡𝐵 ⊆ dom 𝐵)
26	23, 25	eqsstrid 3951	. . 3 ⊢ ((◡𝐵 ⊆ 𝐵 ∧ (𝐵 ∘ ◡𝐵) ⊆ I ) → ran 𝐵 ⊆ dom 𝐵)
27	22, 26	jca 521	. 2 ⊢ ((◡𝐵 ⊆ 𝐵 ∧ (𝐵 ∘ ◡𝐵) ⊆ I ) → ((𝐵 ∘ 𝐵) ⊆ I ∧ ran 𝐵 ⊆ dom 𝐵))
28	13, 27	impbii 212	1 ⊢ (((𝐵 ∘ 𝐵) ⊆ I ∧ ran 𝐵 ⊆ dom 𝐵) ↔ (◡𝐵 ⊆ 𝐵 ∧ (𝐵 ∘ ◡𝐵) ⊆ I ))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ⊆ 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-ne 2960  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:  nvo2  38222