Theorem locfinref 33840

Description: A locally finite refinement of an open cover induces a locally finite open cover with the original index set. This is fact 2 of http://at.yorku.ca/p/a/c/a/02.pdf, it is expressed by exposing a function 𝑓 from the original cover 𝑈, which is taken as the index set. (Contributed by Thierry Arnoux, 31-Jan-2020.)

Hypotheses

Ref	Expression
locfinref.x	⊢ 𝑋 = ∪ 𝐽
locfinref.1	⊢ (𝜑 → 𝑈 ⊆ 𝐽)
locfinref.2	⊢ (𝜑 → 𝑋 = ∪ 𝑈)
locfinref.3	⊢ (𝜑 → 𝑉 ⊆ 𝐽)
locfinref.4	⊢ (𝜑 → 𝑉Ref𝑈)
locfinref.5	⊢ (𝜑 → 𝑉 ∈ (LocFin‘𝐽))

Assertion

Ref	Expression
locfinref	⊢ (𝜑 → ∃𝑓(𝑓:𝑈⟶𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))

Distinct variable groups:   𝑓,𝐽   𝑈,𝑓   𝑓,𝑉   𝜑,𝑓

Allowed substitution hint:   𝑋(𝑓)

Proof of Theorem locfinref

Dummy variables 𝑔 𝑥 𝑛 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f0 6789	. . . 4 ⊢ ∅:∅⟶𝐽
2		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑈 = ∅) → 𝑈 = ∅)
3	2	feq2d 6722	. . . 4 ⊢ ((𝜑 ∧ 𝑈 = ∅) → (∅:𝑈⟶𝐽 ↔ ∅:∅⟶𝐽))
4	1, 3	mpbiri 258	. . 3 ⊢ ((𝜑 ∧ 𝑈 = ∅) → ∅:𝑈⟶𝐽)
5		rn0 5936	. . . . 5 ⊢ ran ∅ = ∅
6		0ex 5307	. . . . . 6 ⊢ ∅ ∈ V
7		refref 23521	. . . . . 6 ⊢ (∅ ∈ V → ∅Ref∅)
8	6, 7	ax-mp 5	. . . . 5 ⊢ ∅Ref∅
9	5, 8	eqbrtri 5164	. . . 4 ⊢ ran ∅Ref∅
10	9, 2	breqtrrid 5181	. . 3 ⊢ ((𝜑 ∧ 𝑈 = ∅) → ran ∅Ref𝑈)
11		sn0top 23006	. . . . . 6 ⊢ {∅} ∈ Top
12	11	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑈 = ∅) → {∅} ∈ Top)
13		eqidd 2738	. . . . 5 ⊢ ((𝜑 ∧ 𝑈 = ∅) → ∅ = ∅)
14		ral0 4513	. . . . . 6 ⊢ ∀𝑥 ∈ ∅ ∃𝑛 ∈ {∅} (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ ran ∅ ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)
15	14	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑈 = ∅) → ∀𝑥 ∈ ∅ ∃𝑛 ∈ {∅} (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ ran ∅ ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
16	6	unisn 4926	. . . . . . 7 ⊢ ∪ {∅} = ∅
17	16	eqcomi 2746	. . . . . 6 ⊢ ∅ = ∪ {∅}
18	5	unieqi 4919	. . . . . . 7 ⊢ ∪ ran ∅ = ∪ ∅
19		uni0 4935	. . . . . . 7 ⊢ ∪ ∅ = ∅
20	18, 19	eqtr2i 2766	. . . . . 6 ⊢ ∅ = ∪ ran ∅
21	17, 20	islocfin 23525	. . . . 5 ⊢ (ran ∅ ∈ (LocFin‘{∅}) ↔ ({∅} ∈ Top ∧ ∅ = ∅ ∧ ∀𝑥 ∈ ∅ ∃𝑛 ∈ {∅} (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ ran ∅ ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
22	12, 13, 15, 21	syl3anbrc 1344	. . . 4 ⊢ ((𝜑 ∧ 𝑈 = ∅) → ran ∅ ∈ (LocFin‘{∅}))
23		locfinref.2	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 = ∪ 𝑈)
24	23	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑈 = ∅) → 𝑋 = ∪ 𝑈)
25	2	unieqd 4920	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑈 = ∅) → ∪ 𝑈 = ∪ ∅)
26	24, 25	eqtrd 2777	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑈 = ∅) → 𝑋 = ∪ ∅)
27		locfinref.x	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
28	26, 27, 19	3eqtr3g 2800	. . . . . 6 ⊢ ((𝜑 ∧ 𝑈 = ∅) → ∪ 𝐽 = ∅)
29		locfinref.5	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ (LocFin‘𝐽))
30		locfintop 23529	. . . . . . . 8 ⊢ (𝑉 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top)
31		0top 22990	. . . . . . . 8 ⊢ (𝐽 ∈ Top → (∪ 𝐽 = ∅ ↔ 𝐽 = {∅}))
32	29, 30, 31	3syl 18	. . . . . . 7 ⊢ (𝜑 → (∪ 𝐽 = ∅ ↔ 𝐽 = {∅}))
33	32	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑈 = ∅) → (∪ 𝐽 = ∅ ↔ 𝐽 = {∅}))
34	28, 33	mpbid 232	. . . . 5 ⊢ ((𝜑 ∧ 𝑈 = ∅) → 𝐽 = {∅})
35	34	fveq2d 6910	. . . 4 ⊢ ((𝜑 ∧ 𝑈 = ∅) → (LocFin‘𝐽) = (LocFin‘{∅}))
36	22, 35	eleqtrrd 2844	. . 3 ⊢ ((𝜑 ∧ 𝑈 = ∅) → ran ∅ ∈ (LocFin‘𝐽))
37		feq1 6716	. . . . 5 ⊢ (𝑓 = ∅ → (𝑓:𝑈⟶𝐽 ↔ ∅:𝑈⟶𝐽))
38		rneq 5947	. . . . . 6 ⊢ (𝑓 = ∅ → ran 𝑓 = ran ∅)
39	38	breq1d 5153	. . . . 5 ⊢ (𝑓 = ∅ → (ran 𝑓Ref𝑈 ↔ ran ∅Ref𝑈))
40	38	eleq1d 2826	. . . . 5 ⊢ (𝑓 = ∅ → (ran 𝑓 ∈ (LocFin‘𝐽) ↔ ran ∅ ∈ (LocFin‘𝐽)))
41	37, 39, 40	3anbi123d 1438	. . . 4 ⊢ (𝑓 = ∅ → ((𝑓:𝑈⟶𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)) ↔ (∅:𝑈⟶𝐽 ∧ ran ∅Ref𝑈 ∧ ran ∅ ∈ (LocFin‘𝐽))))
42	6, 41	spcev 3606	. . 3 ⊢ ((∅:𝑈⟶𝐽 ∧ ran ∅Ref𝑈 ∧ ran ∅ ∈ (LocFin‘𝐽)) → ∃𝑓(𝑓:𝑈⟶𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
43	4, 10, 36, 42	syl3anc 1373	. 2 ⊢ ((𝜑 ∧ 𝑈 = ∅) → ∃𝑓(𝑓:𝑈⟶𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
44		locfinref.1	. . . . 5 ⊢ (𝜑 → 𝑈 ⊆ 𝐽)
45		locfinref.3	. . . . 5 ⊢ (𝜑 → 𝑉 ⊆ 𝐽)
46		locfinref.4	. . . . 5 ⊢ (𝜑 → 𝑉Ref𝑈)
47	27, 44, 23, 45, 46, 29	locfinreflem 33839	. . . 4 ⊢ (𝜑 → ∃𝑔((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽))))
48	47	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑈 ≠ ∅) → ∃𝑔((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽))))
49		simpl 482	. . . 4 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → (𝜑 ∧ 𝑈 ≠ ∅))
50		simprl1 1219	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → Fun 𝑔)
51		fdmrn 6767	. . . . . . . 8 ⊢ (Fun 𝑔 ↔ 𝑔:dom 𝑔⟶ran 𝑔)
52	50, 51	sylib 218	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → 𝑔:dom 𝑔⟶ran 𝑔)
53		simprl3 1221	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ran 𝑔 ⊆ 𝐽)
54	52, 53	fssd 6753	. . . . . 6 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → 𝑔:dom 𝑔⟶𝐽)
55		fconstg 6795	. . . . . . . 8 ⊢ (∅ ∈ V → ((𝑈 ∖ dom 𝑔) × {∅}):(𝑈 ∖ dom 𝑔)⟶{∅})
56	6, 55	mp1i 13	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ((𝑈 ∖ dom 𝑔) × {∅}):(𝑈 ∖ dom 𝑔)⟶{∅})
57		0opn 22910	. . . . . . . . . 10 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽)
58	29, 30, 57	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → ∅ ∈ 𝐽)
59	58	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ∅ ∈ 𝐽)
60	59	snssd 4809	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → {∅} ⊆ 𝐽)
61	56, 60	fssd 6753	. . . . . 6 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ((𝑈 ∖ dom 𝑔) × {∅}):(𝑈 ∖ dom 𝑔)⟶𝐽)
62		disjdif 4472	. . . . . . 7 ⊢ (dom 𝑔 ∩ (𝑈 ∖ dom 𝑔)) = ∅
63	62	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → (dom 𝑔 ∩ (𝑈 ∖ dom 𝑔)) = ∅)
64		fun2 6771	. . . . . 6 ⊢ (((𝑔:dom 𝑔⟶𝐽 ∧ ((𝑈 ∖ dom 𝑔) × {∅}):(𝑈 ∖ dom 𝑔)⟶𝐽) ∧ (dom 𝑔 ∩ (𝑈 ∖ dom 𝑔)) = ∅) → (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})):(dom 𝑔 ∪ (𝑈 ∖ dom 𝑔))⟶𝐽)
65	54, 61, 63, 64	syl21anc 838	. . . . 5 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})):(dom 𝑔 ∪ (𝑈 ∖ dom 𝑔))⟶𝐽)
66		simprl2 1220	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → dom 𝑔 ⊆ 𝑈)
67		undif 4482	. . . . . . 7 ⊢ (dom 𝑔 ⊆ 𝑈 ↔ (dom 𝑔 ∪ (𝑈 ∖ dom 𝑔)) = 𝑈)
68	66, 67	sylib 218	. . . . . 6 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → (dom 𝑔 ∪ (𝑈 ∖ dom 𝑔)) = 𝑈)
69	68	feq2d 6722	. . . . 5 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ((𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})):(dom 𝑔 ∪ (𝑈 ∖ dom 𝑔))⟶𝐽 ↔ (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})):𝑈⟶𝐽))
70	65, 69	mpbid 232	. . . 4 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})):𝑈⟶𝐽)
71		simpr 484	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = ran 𝑔) → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = ran 𝑔)
72		simprrl 781	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ran 𝑔Ref𝑈)
73	72	adantr 480	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = ran 𝑔) → ran 𝑔Ref𝑈)
74	71, 73	eqbrtrd 5165	. . . . 5 ⊢ ((((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = ran 𝑔) → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅}))Ref𝑈)
75		simpr 484	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅})) → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅}))
76	49	simprd 495	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → 𝑈 ≠ ∅)
77		refun0 23523	. . . . . . . 8 ⊢ ((ran 𝑔Ref𝑈 ∧ 𝑈 ≠ ∅) → (ran 𝑔 ∪ {∅})Ref𝑈)
78	72, 76, 77	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → (ran 𝑔 ∪ {∅})Ref𝑈)
79	78	adantr 480	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅})) → (ran 𝑔 ∪ {∅})Ref𝑈)
80	75, 79	eqbrtrd 5165	. . . . 5 ⊢ ((((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅})) → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅}))Ref𝑈)
81		rnxpss 6192	. . . . . . 7 ⊢ ran ((𝑈 ∖ dom 𝑔) × {∅}) ⊆ {∅}
82		sssn 4826	. . . . . . 7 ⊢ (ran ((𝑈 ∖ dom 𝑔) × {∅}) ⊆ {∅} ↔ (ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ ∨ ran ((𝑈 ∖ dom 𝑔) × {∅}) = {∅}))
83	81, 82	mpbi 230	. . . . . 6 ⊢ (ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ ∨ ran ((𝑈 ∖ dom 𝑔) × {∅}) = {∅})
84		rnun 6165	. . . . . . . . 9 ⊢ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ ran ((𝑈 ∖ dom 𝑔) × {∅}))
85		uneq2 4162	. . . . . . . . 9 ⊢ (ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ → (ran 𝑔 ∪ ran ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ ∅))
86	84, 85	eqtrid 2789	. . . . . . . 8 ⊢ (ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ ∅))
87		un0 4394	. . . . . . . 8 ⊢ (ran 𝑔 ∪ ∅) = ran 𝑔
88	86, 87	eqtrdi 2793	. . . . . . 7 ⊢ (ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = ran 𝑔)
89		uneq2 4162	. . . . . . . 8 ⊢ (ran ((𝑈 ∖ dom 𝑔) × {∅}) = {∅} → (ran 𝑔 ∪ ran ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅}))
90	84, 89	eqtrid 2789	. . . . . . 7 ⊢ (ran ((𝑈 ∖ dom 𝑔) × {∅}) = {∅} → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅}))
91	88, 90	orim12i 909	. . . . . 6 ⊢ ((ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ ∨ ran ((𝑈 ∖ dom 𝑔) × {∅}) = {∅}) → (ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = ran 𝑔 ∨ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅})))
92	83, 91	mp1i 13	. . . . 5 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → (ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = ran 𝑔 ∨ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅})))
93	74, 80, 92	mpjaodan 961	. . . 4 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅}))Ref𝑈)
94		simprrr 782	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ran 𝑔 ∈ (LocFin‘𝐽))
95	94	adantr 480	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = ran 𝑔) → ran 𝑔 ∈ (LocFin‘𝐽))
96	71, 95	eqeltrd 2841	. . . . 5 ⊢ ((((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = ran 𝑔) → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ (LocFin‘𝐽))
97	94	adantr 480	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅})) → ran 𝑔 ∈ (LocFin‘𝐽))
98		snfi 9083	. . . . . . . 8 ⊢ {∅} ∈ Fin
99	98	a1i 11	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅})) → {∅} ∈ Fin)
100	59	adantr 480	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅})) → ∅ ∈ 𝐽)
101	100	snssd 4809	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅})) → {∅} ⊆ 𝐽)
102	101	unissd 4917	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅})) → ∪ {∅} ⊆ ∪ 𝐽)
103		lfinun 23533	. . . . . . 7 ⊢ ((ran 𝑔 ∈ (LocFin‘𝐽) ∧ {∅} ∈ Fin ∧ ∪ {∅} ⊆ ∪ 𝐽) → (ran 𝑔 ∪ {∅}) ∈ (LocFin‘𝐽))
104	97, 99, 102, 103	syl3anc 1373	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅})) → (ran 𝑔 ∪ {∅}) ∈ (LocFin‘𝐽))
105	75, 104	eqeltrd 2841	. . . . 5 ⊢ ((((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅})) → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ (LocFin‘𝐽))
106	96, 105, 92	mpjaodan 961	. . . 4 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ (LocFin‘𝐽))
107		refrel 23516	. . . . . . . . 9 ⊢ Rel Ref
108	107	brrelex2i 5742	. . . . . . . 8 ⊢ (𝑉Ref𝑈 → 𝑈 ∈ V)
109		difexg 5329	. . . . . . . 8 ⊢ (𝑈 ∈ V → (𝑈 ∖ dom 𝑔) ∈ V)
110	46, 108, 109	3syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑈 ∖ dom 𝑔) ∈ V)
111	110	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑈 ≠ ∅) → (𝑈 ∖ dom 𝑔) ∈ V)
112		p0ex 5384	. . . . . . 7 ⊢ {∅} ∈ V
113		xpexg 7770	. . . . . . 7 ⊢ (((𝑈 ∖ dom 𝑔) ∈ V ∧ {∅} ∈ V) → ((𝑈 ∖ dom 𝑔) × {∅}) ∈ V)
114	112, 113	mpan2 691	. . . . . 6 ⊢ ((𝑈 ∖ dom 𝑔) ∈ V → ((𝑈 ∖ dom 𝑔) × {∅}) ∈ V)
115		vex 3484	. . . . . . 7 ⊢ 𝑔 ∈ V
116		unexg 7763	. . . . . . 7 ⊢ ((𝑔 ∈ V ∧ ((𝑈 ∖ dom 𝑔) × {∅}) ∈ V) → (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ V)
117	115, 116	mpan 690	. . . . . 6 ⊢ (((𝑈 ∖ dom 𝑔) × {∅}) ∈ V → (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ V)
118		feq1 6716	. . . . . . . 8 ⊢ (𝑓 = (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) → (𝑓:𝑈⟶𝐽 ↔ (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})):𝑈⟶𝐽))
119		rneq 5947	. . . . . . . . 9 ⊢ (𝑓 = (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) → ran 𝑓 = ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})))
120	119	breq1d 5153	. . . . . . . 8 ⊢ (𝑓 = (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) → (ran 𝑓Ref𝑈 ↔ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅}))Ref𝑈))
121	119	eleq1d 2826	. . . . . . . 8 ⊢ (𝑓 = (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) → (ran 𝑓 ∈ (LocFin‘𝐽) ↔ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ (LocFin‘𝐽)))
122	118, 120, 121	3anbi123d 1438	. . . . . . 7 ⊢ (𝑓 = (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) → ((𝑓:𝑈⟶𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)) ↔ ((𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})):𝑈⟶𝐽 ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅}))Ref𝑈 ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ (LocFin‘𝐽))))
123	122	spcegv 3597	. . . . . 6 ⊢ ((𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ V → (((𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})):𝑈⟶𝐽 ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅}))Ref𝑈 ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ (LocFin‘𝐽)) → ∃𝑓(𝑓:𝑈⟶𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))))
124	111, 114, 117, 123	4syl 19	. . . . 5 ⊢ ((𝜑 ∧ 𝑈 ≠ ∅) → (((𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})):𝑈⟶𝐽 ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅}))Ref𝑈 ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ (LocFin‘𝐽)) → ∃𝑓(𝑓:𝑈⟶𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))))
125	124	imp 406	. . . 4 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})):𝑈⟶𝐽 ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅}))Ref𝑈 ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ (LocFin‘𝐽))) → ∃𝑓(𝑓:𝑈⟶𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
126	49, 70, 93, 106, 125	syl13anc 1374	. . 3 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ∃𝑓(𝑓:𝑈⟶𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
127	48, 126	exlimddv 1935	. 2 ⊢ ((𝜑 ∧ 𝑈 ≠ ∅) → ∃𝑓(𝑓:𝑈⟶𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
128	43, 127	pm2.61dane 3029	1 ⊢ (𝜑 → ∃𝑓(𝑓:𝑈⟶𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1540  ∃wex 1779   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  {crab 3436  Vcvv 3480   ∖ cdif 3948   ∪ cun 3949   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333  {csn 4626  ∪ cuni 4907   class class class wbr 5143   × cxp 5683  dom cdm 5685  ran crn 5686  Fun wfun 6555  ⟶wf 6557  ‘cfv 6561  Fincfn 8985  Topctop 22899  Refcref 23510  LocFinclocfin 23512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-reg 9632  ax-inf2 9681  ax-ac2 10503

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-fin 8989  df-r1 9804  df-rank 9805  df-card 9979  df-ac 10156  df-top 22900  df-topon 22917  df-ref 23513  df-locfin 23515

This theorem is referenced by:  pcmplfinf  33860