Theorem cossidcnvss2 38215

Description: Two-class version of cossidcnvss 38216. (Contributed by BJ, 8-Nov-2026.)

Assertion

Ref	Expression
cossidcnvss2	⊢ (((𝐵 ∘ 𝐴) ⊆ I ∧ ran 𝐴 ⊆ dom 𝐵) ↔ (◡𝐴 ⊆ 𝐵 ∧ Fun (𝐵 ↾ ran 𝐴)))

Proof of Theorem cossidcnvss2

Step	Hyp	Ref	Expression
1		bj-cnvresrn 38190	. . . 4 ⊢ (◡𝐴 ↾ ran 𝐴) = ◡𝐴
2		simpr 490	. . . . . 6 ⊢ (((𝐵 ∘ 𝐴) ⊆ I ∧ ran 𝐴 ⊆ dom 𝐵) → ran 𝐴 ⊆ dom 𝐵)
3		ssres2 6009	. . . . . 6 ⊢ (ran 𝐴 ⊆ dom 𝐵 → (◡𝐴 ↾ ran 𝐴) ⊆ (◡𝐴 ↾ dom 𝐵))
4	2, 3	syl 18	. . . . 5 ⊢ (((𝐵 ∘ 𝐴) ⊆ I ∧ ran 𝐴 ⊆ dom 𝐵) → (◡𝐴 ↾ ran 𝐴) ⊆ (◡𝐴 ↾ dom 𝐵))
5		simpl 488	. . . . . 6 ⊢ (((𝐵 ∘ 𝐴) ⊆ I ∧ ran 𝐴 ⊆ dom 𝐵) → (𝐵 ∘ 𝐴) ⊆ I )
6		cossidres2 38214	. . . . . 6 ⊢ ((𝐵 ∘ 𝐴) ⊆ I → (◡𝐴 ↾ dom 𝐵) ⊆ 𝐵)
7	5, 6	syl 18	. . . . 5 ⊢ (((𝐵 ∘ 𝐴) ⊆ I ∧ ran 𝐴 ⊆ dom 𝐵) → (◡𝐴 ↾ dom 𝐵) ⊆ 𝐵)
8	4, 7	sstrd 3923	. . . 4 ⊢ (((𝐵 ∘ 𝐴) ⊆ I ∧ ran 𝐴 ⊆ dom 𝐵) → (◡𝐴 ↾ ran 𝐴) ⊆ 𝐵)
9	1, 8	eqsstrrid 3952	. . 3 ⊢ (((𝐵 ∘ 𝐴) ⊆ I ∧ ran 𝐴 ⊆ dom 𝐵) → ◡𝐴 ⊆ 𝐵)
10		relres 6010	. . . . . 6 ⊢ Rel (𝐵 ↾ ran 𝐴)
11	10	a1i 11	. . . . 5 ⊢ ((𝐵 ∘ 𝐴) ⊆ I → Rel (𝐵 ↾ ran 𝐴))
12		resss 6006	. . . . . . . 8 ⊢ (𝐵 ↾ ran 𝐴) ⊆ 𝐵
13	12	a1i 11	. . . . . . 7 ⊢ ((𝐵 ∘ 𝐴) ⊆ I → (𝐵 ↾ ran 𝐴) ⊆ 𝐵)
14		cossidres1 38213	. . . . . . . . 9 ⊢ ((𝐵 ∘ 𝐴) ⊆ I → (𝐵 ↾ ran 𝐴) ⊆ ◡𝐴)
15		cnvss 5860	. . . . . . . . 9 ⊢ ((𝐵 ↾ ran 𝐴) ⊆ ◡𝐴 → ◡(𝐵 ↾ ran 𝐴) ⊆ ◡◡𝐴)
16	14, 15	syl 18	. . . . . . . 8 ⊢ ((𝐵 ∘ 𝐴) ⊆ I → ◡(𝐵 ↾ ran 𝐴) ⊆ ◡◡𝐴)
17		cnvcnvss 6202	. . . . . . . 8 ⊢ ◡◡𝐴 ⊆ 𝐴
18	16, 17	sstrdi 3925	. . . . . . 7 ⊢ ((𝐵 ∘ 𝐴) ⊆ I → ◡(𝐵 ↾ ran 𝐴) ⊆ 𝐴)
19	13, 18	coss12d 5843	. . . . . 6 ⊢ ((𝐵 ∘ 𝐴) ⊆ I → ((𝐵 ↾ ran 𝐴) ∘ ◡(𝐵 ↾ ran 𝐴)) ⊆ (𝐵 ∘ 𝐴))
20		id 23	. . . . . 6 ⊢ ((𝐵 ∘ 𝐴) ⊆ I → (𝐵 ∘ 𝐴) ⊆ I )
21	19, 20	sstrd 3923	. . . . 5 ⊢ ((𝐵 ∘ 𝐴) ⊆ I → ((𝐵 ↾ ran 𝐴) ∘ ◡(𝐵 ↾ ran 𝐴)) ⊆ I )
22		df-fun 6566	. . . . 5 ⊢ (Fun (𝐵 ↾ ran 𝐴) ↔ (Rel (𝐵 ↾ ran 𝐴) ∧ ((𝐵 ↾ ran 𝐴) ∘ ◡(𝐵 ↾ ran 𝐴)) ⊆ I ))
23	11, 21, 22	sylanbrc 595	. . . 4 ⊢ ((𝐵 ∘ 𝐴) ⊆ I → Fun (𝐵 ↾ ran 𝐴))
24	5, 23	syl 18	. . 3 ⊢ (((𝐵 ∘ 𝐴) ⊆ I ∧ ran 𝐴 ⊆ dom 𝐵) → Fun (𝐵 ↾ ran 𝐴))
25	9, 24	jca 521	. 2 ⊢ (((𝐵 ∘ 𝐴) ⊆ I ∧ ran 𝐴 ⊆ dom 𝐵) → (◡𝐴 ⊆ 𝐵 ∧ Fun (𝐵 ↾ ran 𝐴)))
26		coresrn 38197	. . . 4 ⊢ ((𝐵 ↾ ran 𝐴) ∘ 𝐴) = (𝐵 ∘ 𝐴)
27		cocnvcnv2 6277	. . . . 5 ⊢ ((𝐵 ↾ ran 𝐴) ∘ ◡◡𝐴) = ((𝐵 ↾ ran 𝐴) ∘ 𝐴)
28		simpl 488	. . . . . . . . . 10 ⊢ ((◡𝐴 ⊆ 𝐵 ∧ Fun (𝐵 ↾ ran 𝐴)) → ◡𝐴 ⊆ 𝐵)
29		ssres 6008	. . . . . . . . . 10 ⊢ (◡𝐴 ⊆ 𝐵 → (◡𝐴 ↾ ran 𝐴) ⊆ (𝐵 ↾ ran 𝐴))
30	28, 29	syl 18	. . . . . . . . 9 ⊢ ((◡𝐴 ⊆ 𝐵 ∧ Fun (𝐵 ↾ ran 𝐴)) → (◡𝐴 ↾ ran 𝐴) ⊆ (𝐵 ↾ ran 𝐴))
31	1, 30	eqsstrrid 3952	. . . . . . . 8 ⊢ ((◡𝐴 ⊆ 𝐵 ∧ Fun (𝐵 ↾ ran 𝐴)) → ◡𝐴 ⊆ (𝐵 ↾ ran 𝐴))
32		cnvss 5860	. . . . . . . 8 ⊢ (◡𝐴 ⊆ (𝐵 ↾ ran 𝐴) → ◡◡𝐴 ⊆ ◡(𝐵 ↾ ran 𝐴))
33	31, 32	syl 18	. . . . . . 7 ⊢ ((◡𝐴 ⊆ 𝐵 ∧ Fun (𝐵 ↾ ran 𝐴)) → ◡◡𝐴 ⊆ ◡(𝐵 ↾ ran 𝐴))
34		coss2 5842	. . . . . . 7 ⊢ (◡◡𝐴 ⊆ ◡(𝐵 ↾ ran 𝐴) → ((𝐵 ↾ ran 𝐴) ∘ ◡◡𝐴) ⊆ ((𝐵 ↾ ran 𝐴) ∘ ◡(𝐵 ↾ ran 𝐴)))
35	33, 34	syl 18	. . . . . 6 ⊢ ((◡𝐴 ⊆ 𝐵 ∧ Fun (𝐵 ↾ ran 𝐴)) → ((𝐵 ↾ ran 𝐴) ∘ ◡◡𝐴) ⊆ ((𝐵 ↾ ran 𝐴) ∘ ◡(𝐵 ↾ ran 𝐴)))
36		simpr 490	. . . . . . 7 ⊢ ((◡𝐴 ⊆ 𝐵 ∧ Fun (𝐵 ↾ ran 𝐴)) → Fun (𝐵 ↾ ran 𝐴))
37	22	simprbi 503	. . . . . . 7 ⊢ (Fun (𝐵 ↾ ran 𝐴) → ((𝐵 ↾ ran 𝐴) ∘ ◡(𝐵 ↾ ran 𝐴)) ⊆ I )
38	36, 37	syl 18	. . . . . 6 ⊢ ((◡𝐴 ⊆ 𝐵 ∧ Fun (𝐵 ↾ ran 𝐴)) → ((𝐵 ↾ ran 𝐴) ∘ ◡(𝐵 ↾ ran 𝐴)) ⊆ I )
39	35, 38	sstrd 3923	. . . . 5 ⊢ ((◡𝐴 ⊆ 𝐵 ∧ Fun (𝐵 ↾ ran 𝐴)) → ((𝐵 ↾ ran 𝐴) ∘ ◡◡𝐴) ⊆ I )
40	27, 39	eqsstrrid 3952	. . . 4 ⊢ ((◡𝐴 ⊆ 𝐵 ∧ Fun (𝐵 ↾ ran 𝐴)) → ((𝐵 ↾ ran 𝐴) ∘ 𝐴) ⊆ I )
41	26, 40	eqsstrrid 3952	. . 3 ⊢ ((◡𝐴 ⊆ 𝐵 ∧ Fun (𝐵 ↾ ran 𝐴)) → (𝐵 ∘ 𝐴) ⊆ I )
42		df-rn 5667	. . . 4 ⊢ ran 𝐴 = dom ◡𝐴
43		dmss 5894	. . . . 5 ⊢ (◡𝐴 ⊆ 𝐵 → dom ◡𝐴 ⊆ dom 𝐵)
44	28, 43	syl 18	. . . 4 ⊢ ((◡𝐴 ⊆ 𝐵 ∧ Fun (𝐵 ↾ ran 𝐴)) → dom ◡𝐴 ⊆ dom 𝐵)
45	42, 44	eqsstrid 3951	. . 3 ⊢ ((◡𝐴 ⊆ 𝐵 ∧ Fun (𝐵 ↾ ran 𝐴)) → ran 𝐴 ⊆ dom 𝐵)
46	41, 45	jca 521	. 2 ⊢ ((◡𝐴 ⊆ 𝐵 ∧ Fun (𝐵 ↾ ran 𝐴)) → ((𝐵 ∘ 𝐴) ⊆ I ∧ ran 𝐴 ⊆ dom 𝐵))
47	25, 46	impbii 212	1 ⊢ (((𝐵 ∘ 𝐴) ⊆ I ∧ ran 𝐴 ⊆ dom 𝐵) ↔ (◡𝐴 ⊆ 𝐵 ∧ Fun (𝐵 ↾ ran 𝐴)))

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  Rel wrel 5661  Fun wfun 6558

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  df-fun 6566

This theorem is used by: (None)