Theorem nvo3 38223

Description: A class is sub-involutive if and only if its restriction to the intersection of its domain and range is a bijection of that intersection onto itself and coincides with the restriction of its converse. (Contributed by BJ, 8-Nov-2026.)

Hypothesis

Ref	Expression
nvo3.def	⊢ 𝑋 = (dom 𝐴 ∩ ran 𝐴)

Assertion

Ref	Expression
nvo3	⊢ ((𝐴 ∘ 𝐴) ⊆ I ↔ ((𝐴 ↾ 𝑋):𝑋⟶1-1onto𝑋 ∧ (◡𝐴 ↾ 𝑋) = (𝐴 ↾ 𝑋)))

Proof of Theorem nvo3

Step	Hyp	Ref	Expression
1		relres 6010	. . . . . . . . 9 ⊢ Rel (𝐴 ↾ 𝑋)
2	1	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → Rel (𝐴 ↾ 𝑋))
3		resss 6006	. . . . . . . . . . 11 ⊢ (𝐴 ↾ 𝑋) ⊆ 𝐴
4	3	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → (𝐴 ↾ 𝑋) ⊆ 𝐴)
5		nvo3.def	. . . . . . . . . . . . . . 15 ⊢ 𝑋 = (dom 𝐴 ∩ ran 𝐴)
6		inss2 4137	. . . . . . . . . . . . . . 15 ⊢ (dom 𝐴 ∩ ran 𝐴) ⊆ ran 𝐴
7	5, 6	eqsstri 3959	. . . . . . . . . . . . . 14 ⊢ 𝑋 ⊆ ran 𝐴
8		ssres2 6009	. . . . . . . . . . . . . 14 ⊢ (𝑋 ⊆ ran 𝐴 → (𝐴 ↾ 𝑋) ⊆ (𝐴 ↾ ran 𝐴))
9	7, 8	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (𝐴 ↾ 𝑋) ⊆ (𝐴 ↾ ran 𝐴)
10		cossidres1 38213	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → (𝐴 ↾ ran 𝐴) ⊆ ◡𝐴)
11	9, 10	sstrid 3924	. . . . . . . . . . . 12 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → (𝐴 ↾ 𝑋) ⊆ ◡𝐴)
12		cnvss 5860	. . . . . . . . . . . 12 ⊢ ((𝐴 ↾ 𝑋) ⊆ ◡𝐴 → ◡(𝐴 ↾ 𝑋) ⊆ ◡◡𝐴)
13	11, 12	syl 18	. . . . . . . . . . 11 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → ◡(𝐴 ↾ 𝑋) ⊆ ◡◡𝐴)
14		cnvcnvss 6202	. . . . . . . . . . 11 ⊢ ◡◡𝐴 ⊆ 𝐴
15	13, 14	sstrdi 3925	. . . . . . . . . 10 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → ◡(𝐴 ↾ 𝑋) ⊆ 𝐴)
16	4, 15	coss12d 5843	. . . . . . . . 9 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → ((𝐴 ↾ 𝑋) ∘ ◡(𝐴 ↾ 𝑋)) ⊆ (𝐴 ∘ 𝐴))
17		id 23	. . . . . . . . 9 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → (𝐴 ∘ 𝐴) ⊆ I )
18	16, 17	sstrd 3923	. . . . . . . 8 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → ((𝐴 ↾ 𝑋) ∘ ◡(𝐴 ↾ 𝑋)) ⊆ I )
19		df-fun 6566	. . . . . . . 8 ⊢ (Fun (𝐴 ↾ 𝑋) ↔ (Rel (𝐴 ↾ 𝑋) ∧ ((𝐴 ↾ 𝑋) ∘ ◡(𝐴 ↾ 𝑋)) ⊆ I ))
20	2, 18, 19	sylanbrc 595	. . . . . . 7 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → Fun (𝐴 ↾ 𝑋))
21		inss1 4136	. . . . . . . . . 10 ⊢ (dom 𝐴 ∩ ran 𝐴) ⊆ dom 𝐴
22	5, 21	eqsstri 3959	. . . . . . . . 9 ⊢ 𝑋 ⊆ dom 𝐴
23		ssdmres 6019	. . . . . . . . 9 ⊢ (𝑋 ⊆ dom 𝐴 ↔ dom (𝐴 ↾ 𝑋) = 𝑋)
24	22, 23	mpbi 233	. . . . . . . 8 ⊢ dom (𝐴 ↾ 𝑋) = 𝑋
25	24	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → dom (𝐴 ↾ 𝑋) = 𝑋)
26		df-fn 6567	. . . . . . 7 ⊢ ((𝐴 ↾ 𝑋) Fn 𝑋 ↔ (Fun (𝐴 ↾ 𝑋) ∧ dom (𝐴 ↾ 𝑋) = 𝑋))
27	20, 25, 26	sylanbrc 595	. . . . . 6 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → (𝐴 ↾ 𝑋) Fn 𝑋)
28		rnss 5931	. . . . . . . . . 10 ⊢ ((𝐴 ↾ 𝑋) ⊆ ◡𝐴 → ran (𝐴 ↾ 𝑋) ⊆ ran ◡𝐴)
29	11, 28	syl 18	. . . . . . . . 9 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → ran (𝐴 ↾ 𝑋) ⊆ ran ◡𝐴)
30		dfdm4 5887	. . . . . . . . 9 ⊢ dom 𝐴 = ran ◡𝐴
31	29, 30	sseqtrrdi 3954	. . . . . . . 8 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → ran (𝐴 ↾ 𝑋) ⊆ dom 𝐴)
32		rnresss 6021	. . . . . . . . 9 ⊢ ran (𝐴 ↾ 𝑋) ⊆ ran 𝐴
33	32	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → ran (𝐴 ↾ 𝑋) ⊆ ran 𝐴)
34	31, 33	ssind 4140	. . . . . . 7 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → ran (𝐴 ↾ 𝑋) ⊆ (dom 𝐴 ∩ ran 𝐴))
35	34, 5	sseqtrrdi 3954	. . . . . 6 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → ran (𝐴 ↾ 𝑋) ⊆ 𝑋)
36		df-f 6568	. . . . . 6 ⊢ ((𝐴 ↾ 𝑋):𝑋⟶𝑋 ↔ ((𝐴 ↾ 𝑋) Fn 𝑋 ∧ ran (𝐴 ↾ 𝑋) ⊆ 𝑋))
37	27, 35, 36	sylanbrc 595	. . . . 5 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → (𝐴 ↾ 𝑋):𝑋⟶𝑋)
38		ffn 6737	. . . . 5 ⊢ ((𝐴 ↾ 𝑋):𝑋⟶𝑋 → (𝐴 ↾ 𝑋) Fn 𝑋)
39	37, 38	syl 18	. . . 4 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → (𝐴 ↾ 𝑋) Fn 𝑋)
40	4, 4	coss12d 5843	. . . . . 6 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → ((𝐴 ↾ 𝑋) ∘ (𝐴 ↾ 𝑋)) ⊆ (𝐴 ∘ 𝐴))
41	40, 17	sstrd 3923	. . . . 5 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → ((𝐴 ↾ 𝑋) ∘ (𝐴 ↾ 𝑋)) ⊆ I )
42		nvo2 38222	. . . . . 6 ⊢ ((𝐴 ↾ 𝑋):𝑋⟶𝑋 → (((𝐴 ↾ 𝑋) ∘ (𝐴 ↾ 𝑋)) ⊆ I ↔ ◡(𝐴 ↾ 𝑋) = (𝐴 ↾ 𝑋)))
43	37, 42	syl 18	. . . . 5 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → (((𝐴 ↾ 𝑋) ∘ (𝐴 ↾ 𝑋)) ⊆ I ↔ ◡(𝐴 ↾ 𝑋) = (𝐴 ↾ 𝑋)))
44	41, 43	mpbid 235	. . . 4 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → ◡(𝐴 ↾ 𝑋) = (𝐴 ↾ 𝑋))
45		nvof1o 7317	. . . 4 ⊢ (((𝐴 ↾ 𝑋) Fn 𝑋 ∧ ◡(𝐴 ↾ 𝑋) = (𝐴 ↾ 𝑋)) → (𝐴 ↾ 𝑋):𝑋⟶1-1onto𝑋)
46	39, 44, 45	syl2anc 596	. . 3 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → (𝐴 ↾ 𝑋):𝑋⟶1-1onto𝑋)
47		residm 6016	. . . . 5 ⊢ ((◡𝐴 ↾ 𝑋) ↾ 𝑋) = (◡𝐴 ↾ 𝑋)
48		ssres2 6009	. . . . . . . 8 ⊢ (𝑋 ⊆ dom 𝐴 → (◡𝐴 ↾ 𝑋) ⊆ (◡𝐴 ↾ dom 𝐴))
49	22, 48	ax-mp 5	. . . . . . 7 ⊢ (◡𝐴 ↾ 𝑋) ⊆ (◡𝐴 ↾ dom 𝐴)
50		cossidres2 38214	. . . . . . 7 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → (◡𝐴 ↾ dom 𝐴) ⊆ 𝐴)
51	49, 50	sstrid 3924	. . . . . 6 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → (◡𝐴 ↾ 𝑋) ⊆ 𝐴)
52		ssres 6008	. . . . . 6 ⊢ ((◡𝐴 ↾ 𝑋) ⊆ 𝐴 → ((◡𝐴 ↾ 𝑋) ↾ 𝑋) ⊆ (𝐴 ↾ 𝑋))
53	51, 52	syl 18	. . . . 5 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → ((◡𝐴 ↾ 𝑋) ↾ 𝑋) ⊆ (𝐴 ↾ 𝑋))
54	47, 53	eqsstrrid 3952	. . . 4 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → (◡𝐴 ↾ 𝑋) ⊆ (𝐴 ↾ 𝑋))
55		residm 6016	. . . . 5 ⊢ ((𝐴 ↾ 𝑋) ↾ 𝑋) = (𝐴 ↾ 𝑋)
56		ssres 6008	. . . . . 6 ⊢ ((𝐴 ↾ 𝑋) ⊆ ◡𝐴 → ((𝐴 ↾ 𝑋) ↾ 𝑋) ⊆ (◡𝐴 ↾ 𝑋))
57	11, 56	syl 18	. . . . 5 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → ((𝐴 ↾ 𝑋) ↾ 𝑋) ⊆ (◡𝐴 ↾ 𝑋))
58	55, 57	eqsstrrid 3952	. . . 4 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → (𝐴 ↾ 𝑋) ⊆ (◡𝐴 ↾ 𝑋))
59	54, 58	eqssd 3930	. . 3 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → (◡𝐴 ↾ 𝑋) = (𝐴 ↾ 𝑋))
60	46, 59	jca 521	. 2 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → ((𝐴 ↾ 𝑋):𝑋⟶1-1onto𝑋 ∧ (◡𝐴 ↾ 𝑋) = (𝐴 ↾ 𝑋)))
61		dmres 6018	. . . . . . 7 ⊢ dom (𝐴 ↾ 𝑋) = (𝑋 ∩ dom 𝐴)
62		inss1 4136	. . . . . . 7 ⊢ (𝑋 ∩ dom 𝐴) ⊆ 𝑋
63	61, 62	eqsstri 3959	. . . . . 6 ⊢ dom (𝐴 ↾ 𝑋) ⊆ 𝑋
64		cores2 6278	. . . . . 6 ⊢ (dom (𝐴 ↾ 𝑋) ⊆ 𝑋 → ((𝐴 ↾ 𝑋) ∘ ◡(◡𝐴 ↾ 𝑋)) = ((𝐴 ↾ 𝑋) ∘ 𝐴))
65	63, 64	ax-mp 5	. . . . 5 ⊢ ((𝐴 ↾ 𝑋) ∘ ◡(◡𝐴 ↾ 𝑋)) = ((𝐴 ↾ 𝑋) ∘ 𝐴)
66		incom 4109	. . . . . . . . . 10 ⊢ (dom 𝐴 ∩ ran 𝐴) = (ran 𝐴 ∩ dom 𝐴)
67	5, 66	eqtri 2786	. . . . . . . . 9 ⊢ 𝑋 = (ran 𝐴 ∩ dom 𝐴)
68	67	reseq2i 5981	. . . . . . . 8 ⊢ (𝐴 ↾ 𝑋) = (𝐴 ↾ (ran 𝐴 ∩ dom 𝐴))
69		resindm 6035	. . . . . . . 8 ⊢ (𝐴 ↾ (ran 𝐴 ∩ dom 𝐴)) = (𝐴 ↾ ran 𝐴)
70	68, 69	eqtri 2786	. . . . . . 7 ⊢ (𝐴 ↾ 𝑋) = (𝐴 ↾ ran 𝐴)
71	70	coeq1i 5846	. . . . . 6 ⊢ ((𝐴 ↾ 𝑋) ∘ 𝐴) = ((𝐴 ↾ ran 𝐴) ∘ 𝐴)
72		coresrn 38197	. . . . . 6 ⊢ ((𝐴 ↾ ran 𝐴) ∘ 𝐴) = (𝐴 ∘ 𝐴)
73	71, 72	eqtri 2786	. . . . 5 ⊢ ((𝐴 ↾ 𝑋) ∘ 𝐴) = (𝐴 ∘ 𝐴)
74	65, 73	eqtri 2786	. . . 4 ⊢ ((𝐴 ↾ 𝑋) ∘ ◡(◡𝐴 ↾ 𝑋)) = (𝐴 ∘ 𝐴)
75		simpr 490	. . . . . . 7 ⊢ (((𝐴 ↾ 𝑋):𝑋⟶1-1onto𝑋 ∧ (◡𝐴 ↾ 𝑋) = (𝐴 ↾ 𝑋)) → (◡𝐴 ↾ 𝑋) = (𝐴 ↾ 𝑋))
76	75	cnveqd 5863	. . . . . 6 ⊢ (((𝐴 ↾ 𝑋):𝑋⟶1-1onto𝑋 ∧ (◡𝐴 ↾ 𝑋) = (𝐴 ↾ 𝑋)) → ◡(◡𝐴 ↾ 𝑋) = ◡(𝐴 ↾ 𝑋))
77	76	coeq2d 5849	. . . . 5 ⊢ (((𝐴 ↾ 𝑋):𝑋⟶1-1onto𝑋 ∧ (◡𝐴 ↾ 𝑋) = (𝐴 ↾ 𝑋)) → ((𝐴 ↾ 𝑋) ∘ ◡(◡𝐴 ↾ 𝑋)) = ((𝐴 ↾ 𝑋) ∘ ◡(𝐴 ↾ 𝑋)))
78		simpl 488	. . . . . 6 ⊢ (((𝐴 ↾ 𝑋):𝑋⟶1-1onto𝑋 ∧ (◡𝐴 ↾ 𝑋) = (𝐴 ↾ 𝑋)) → (𝐴 ↾ 𝑋):𝑋⟶1-1onto𝑋)
79		f1ococnv2 6881	. . . . . 6 ⊢ ((𝐴 ↾ 𝑋):𝑋⟶1-1onto𝑋 → ((𝐴 ↾ 𝑋) ∘ ◡(𝐴 ↾ 𝑋)) = ( I ↾ 𝑋))
80	78, 79	syl 18	. . . . 5 ⊢ (((𝐴 ↾ 𝑋):𝑋⟶1-1onto𝑋 ∧ (◡𝐴 ↾ 𝑋) = (𝐴 ↾ 𝑋)) → ((𝐴 ↾ 𝑋) ∘ ◡(𝐴 ↾ 𝑋)) = ( I ↾ 𝑋))
81	77, 80	eqtrd 2798	. . . 4 ⊢ (((𝐴 ↾ 𝑋):𝑋⟶1-1onto𝑋 ∧ (◡𝐴 ↾ 𝑋) = (𝐴 ↾ 𝑋)) → ((𝐴 ↾ 𝑋) ∘ ◡(◡𝐴 ↾ 𝑋)) = ( I ↾ 𝑋))
82	74, 81	eqtr3id 2812	. . 3 ⊢ (((𝐴 ↾ 𝑋):𝑋⟶1-1onto𝑋 ∧ (◡𝐴 ↾ 𝑋) = (𝐴 ↾ 𝑋)) → (𝐴 ∘ 𝐴) = ( I ↾ 𝑋))
83		resss 6006	. . 3 ⊢ ( I ↾ 𝑋) ⊆ I
84	82, 83	eqsstrdi 3957	. 2 ⊢ (((𝐴 ↾ 𝑋):𝑋⟶1-1onto𝑋 ∧ (◡𝐴 ↾ 𝑋) = (𝐴 ↾ 𝑋)) → (𝐴 ∘ 𝐴) ⊆ I )
85	60, 84	impbii 212	1 ⊢ ((𝐴 ∘ 𝐴) ⊆ I ↔ ((𝐴 ↾ 𝑋):𝑋⟶1-1onto𝑋 ∧ (◡𝐴 ↾ 𝑋) = (𝐴 ↾ 𝑋)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ⊆ wss 3900   ∩ cin 4092   I cid 5550  ^◡ccnv 5655  dom cdm 5656  ran crn 5657   ↾ cres 5658   ∘ ccom 5660  Rel wrel 5661  Fun wfun 6558   Fn wfn 6559  ⟶wf 6560  ⟶1-1ontowf1o 6563

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  df-fn 6567  df-f 6568  df-f1 6569  df-fo 6570  df-f1o 6571

This theorem is used by: (None)