Theorem locfinref 30232

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 6297	. . . 4 ⊢ ∅:∅⟶𝐽
2		simpr 473	. . . . 5 ⊢ ((𝜑 ∧ 𝑈 = ∅) → 𝑈 = ∅)
3	2	feq2d 6238	. . . 4 ⊢ ((𝜑 ∧ 𝑈 = ∅) → (∅:𝑈⟶𝐽 ↔ ∅:∅⟶𝐽))
4	1, 3	mpbiri 249	. . 3 ⊢ ((𝜑 ∧ 𝑈 = ∅) → ∅:𝑈⟶𝐽)
5		rn0 5578	. . . . 5 ⊢ ran ∅ = ∅
6		0ex 4984	. . . . . 6 ⊢ ∅ ∈ V
7		refref 21527	. . . . . 6 ⊢ (∅ ∈ V → ∅Ref∅)
8	6, 7	ax-mp 5	. . . . 5 ⊢ ∅Ref∅
9	5, 8	eqbrtri 4865	. . . 4 ⊢ ran ∅Ref∅
10	9, 2	syl5breqr 4882	. . 3 ⊢ ((𝜑 ∧ 𝑈 = ∅) → ran ∅Ref𝑈)
11		sn0top 21014	. . . . . 6 ⊢ {∅} ∈ Top
12	11	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑈 = ∅) → {∅} ∈ Top)
13		eqidd 2807	. . . . 5 ⊢ ((𝜑 ∧ 𝑈 = ∅) → ∅ = ∅)
14		ral0 4271	. . . . . 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 4646	. . . . . . 7 ⊢ ∪ {∅} = ∅
17	16	eqcomi 2815	. . . . . 6 ⊢ ∅ = ∪ {∅}
18	5	unieqi 4639	. . . . . . 7 ⊢ ∪ ran ∅ = ∪ ∅
19		uni0 4659	. . . . . . 7 ⊢ ∪ ∅ = ∅
20	18, 19	eqtr2i 2829	. . . . . 6 ⊢ ∅ = ∪ ran ∅
21	17, 20	islocfin 21531	. . . . 5 ⊢ (ran ∅ ∈ (LocFin‘{∅}) ↔ ({∅} ∈ Top ∧ ∅ = ∅ ∧ ∀𝑥 ∈ ∅ ∃𝑛 ∈ {∅} (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ ran ∅ ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
22	12, 13, 15, 21	syl3anbrc 1436	. . . 4 ⊢ ((𝜑 ∧ 𝑈 = ∅) → ran ∅ ∈ (LocFin‘{∅}))
23		locfinref.2	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 = ∪ 𝑈)
24	23	adantr 468	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑈 = ∅) → 𝑋 = ∪ 𝑈)
25	2	unieqd 4640	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑈 = ∅) → ∪ 𝑈 = ∪ ∅)
26	24, 25	eqtrd 2840	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑈 = ∅) → 𝑋 = ∪ ∅)
27		locfinref.x	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
28	26, 27, 19	3eqtr3g 2863	. . . . . 6 ⊢ ((𝜑 ∧ 𝑈 = ∅) → ∪ 𝐽 = ∅)
29		locfinref.5	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ (LocFin‘𝐽))
30		locfintop 21535	. . . . . . . 8 ⊢ (𝑉 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top)
31		0top 20998	. . . . . . . 8 ⊢ (𝐽 ∈ Top → (∪ 𝐽 = ∅ ↔ 𝐽 = {∅}))
32	29, 30, 31	3syl 18	. . . . . . 7 ⊢ (𝜑 → (∪ 𝐽 = ∅ ↔ 𝐽 = {∅}))
33	32	adantr 468	. . . . . 6 ⊢ ((𝜑 ∧ 𝑈 = ∅) → (∪ 𝐽 = ∅ ↔ 𝐽 = {∅}))
34	28, 33	mpbid 223	. . . . 5 ⊢ ((𝜑 ∧ 𝑈 = ∅) → 𝐽 = {∅})
35	34	fveq2d 6408	. . . 4 ⊢ ((𝜑 ∧ 𝑈 = ∅) → (LocFin‘𝐽) = (LocFin‘{∅}))
36	22, 35	eleqtrrd 2888	. . 3 ⊢ ((𝜑 ∧ 𝑈 = ∅) → ran ∅ ∈ (LocFin‘𝐽))
37		feq1 6233	. . . . 5 ⊢ (𝑓 = ∅ → (𝑓:𝑈⟶𝐽 ↔ ∅:𝑈⟶𝐽))
38		rneq 5552	. . . . . 6 ⊢ (𝑓 = ∅ → ran 𝑓 = ran ∅)
39	38	breq1d 4854	. . . . 5 ⊢ (𝑓 = ∅ → (ran 𝑓Ref𝑈 ↔ ran ∅Ref𝑈))
40	38	eleq1d 2870	. . . . 5 ⊢ (𝑓 = ∅ → (ran 𝑓 ∈ (LocFin‘𝐽) ↔ ran ∅ ∈ (LocFin‘𝐽)))
41	37, 39, 40	3anbi123d 1553	. . . 4 ⊢ (𝑓 = ∅ → ((𝑓:𝑈⟶𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)) ↔ (∅:𝑈⟶𝐽 ∧ ran ∅Ref𝑈 ∧ ran ∅ ∈ (LocFin‘𝐽))))
42	6, 41	spcev 3493	. . 3 ⊢ ((∅:𝑈⟶𝐽 ∧ ran ∅Ref𝑈 ∧ ran ∅ ∈ (LocFin‘𝐽)) → ∃𝑓(𝑓:𝑈⟶𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
43	4, 10, 36, 42	syl3anc 1483	. 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 30231	. . . 4 ⊢ (𝜑 → ∃𝑔((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽))))
48	47	adantr 468	. . 3 ⊢ ((𝜑 ∧ 𝑈 ≠ ∅) → ∃𝑔((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽))))
49		simpl 470	. . . 4 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → (𝜑 ∧ 𝑈 ≠ ∅))
50		simprl1 1274	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → Fun 𝑔)
51		fdmrn 6275	. . . . . . . 8 ⊢ (Fun 𝑔 ↔ 𝑔:dom 𝑔⟶ran 𝑔)
52	50, 51	sylib 209	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → 𝑔:dom 𝑔⟶ran 𝑔)
53		simprl3 1278	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ran 𝑔 ⊆ 𝐽)
54	52, 53	fssd 6266	. . . . . 6 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → 𝑔:dom 𝑔⟶𝐽)
55		fconstg 6303	. . . . . . . 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 20919	. . . . . . . . . 10 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽)
58	29, 30, 57	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → ∅ ∈ 𝐽)
59	58	ad2antrr 708	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ∅ ∈ 𝐽)
60	59	snssd 4530	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → {∅} ⊆ 𝐽)
61	56, 60	fssd 6266	. . . . . 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 4236	. . . . . . 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 6278	. . . . . 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 857	. . . . 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 1276	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → dom 𝑔 ⊆ 𝑈)
67		undif 4245	. . . . . . 7 ⊢ (dom 𝑔 ⊆ 𝑈 ↔ (dom 𝑔 ∪ (𝑈 ∖ dom 𝑔)) = 𝑈)
68	66, 67	sylib 209	. . . . . 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 6238	. . . . 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 223	. . . 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 473	. . . . . 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 790	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ran 𝑔Ref𝑈)
73	72	adantr 468	. . . . . 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 4866	. . . . 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 473	. . . . . 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 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → 𝑈 ≠ ∅)
77		refun0 21529	. . . . . . . 8 ⊢ ((ran 𝑔Ref𝑈 ∧ 𝑈 ≠ ∅) → (ran 𝑔 ∪ {∅})Ref𝑈)
78	72, 76, 77	syl2anc 575	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → (ran 𝑔 ∪ {∅})Ref𝑈)
79	78	adantr 468	. . . . . 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 4866	. . . . 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 5777	. . . . . . 7 ⊢ ran ((𝑈 ∖ dom 𝑔) × {∅}) ⊆ {∅}
82		sssn 4547	. . . . . . 7 ⊢ (ran ((𝑈 ∖ dom 𝑔) × {∅}) ⊆ {∅} ↔ (ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ ∨ ran ((𝑈 ∖ dom 𝑔) × {∅}) = {∅}))
83	81, 82	mpbi 221	. . . . . 6 ⊢ (ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ ∨ ran ((𝑈 ∖ dom 𝑔) × {∅}) = {∅})
84		rnun 5752	. . . . . . . . 9 ⊢ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ ran ((𝑈 ∖ dom 𝑔) × {∅}))
85		uneq2 3960	. . . . . . . . 9 ⊢ (ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ → (ran 𝑔 ∪ ran ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ ∅))
86	84, 85	syl5eq 2852	. . . . . . . 8 ⊢ (ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ ∅))
87		un0 4165	. . . . . . . 8 ⊢ (ran 𝑔 ∪ ∅) = ran 𝑔
88	86, 87	syl6eq 2856	. . . . . . 7 ⊢ (ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = ran 𝑔)
89		uneq2 3960	. . . . . . . 8 ⊢ (ran ((𝑈 ∖ dom 𝑔) × {∅}) = {∅} → (ran 𝑔 ∪ ran ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅}))
90	84, 89	syl5eq 2852	. . . . . . 7 ⊢ (ran ((𝑈 ∖ dom 𝑔) × {∅}) = {∅} → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅}))
91	88, 90	orim12i 923	. . . . . 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 972	. . . 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 791	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔 ⊆ 𝑈 ∧ ran 𝑔 ⊆ 𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ran 𝑔 ∈ (LocFin‘𝐽))
95	94	adantr 468	. . . . . 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 2885	. . . . 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 468	. . . . . . 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 8273	. . . . . . . 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 468	. . . . . . . . 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 4530	. . . . . . . 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 4656	. . . . . . 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 21539	. . . . . . 7 ⊢ ((ran 𝑔 ∈ (LocFin‘𝐽) ∧ {∅} ∈ Fin ∧ ∪ {∅} ⊆ ∪ 𝐽) → (ran 𝑔 ∪ {∅}) ∈ (LocFin‘𝐽))
104	97, 99, 102, 103	syl3anc 1483	. . . . . 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 2885	. . . . 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 972	. . . 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 21522	. . . . . . . . 9 ⊢ Rel Ref
108	107	brrelex2i 5359	. . . . . . . 8 ⊢ (𝑉Ref𝑈 → 𝑈 ∈ V)
109		difexg 5003	. . . . . . . 8 ⊢ (𝑈 ∈ V → (𝑈 ∖ dom 𝑔) ∈ V)
110	46, 108, 109	3syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑈 ∖ dom 𝑔) ∈ V)
111	110	adantr 468	. . . . . 6 ⊢ ((𝜑 ∧ 𝑈 ≠ ∅) → (𝑈 ∖ dom 𝑔) ∈ V)
112		p0ex 5053	. . . . . . 7 ⊢ {∅} ∈ V
113		xpexg 7186	. . . . . . 7 ⊢ (((𝑈 ∖ dom 𝑔) ∈ V ∧ {∅} ∈ V) → ((𝑈 ∖ dom 𝑔) × {∅}) ∈ V)
114	112, 113	mpan2 674	. . . . . 6 ⊢ ((𝑈 ∖ dom 𝑔) ∈ V → ((𝑈 ∖ dom 𝑔) × {∅}) ∈ V)
115		vex 3394	. . . . . . 7 ⊢ 𝑔 ∈ V
116		unexg 7185	. . . . . . 7 ⊢ ((𝑔 ∈ V ∧ ((𝑈 ∖ dom 𝑔) × {∅}) ∈ V) → (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ V)
117	115, 116	mpan 673	. . . . . 6 ⊢ (((𝑈 ∖ dom 𝑔) × {∅}) ∈ V → (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ V)
118		feq1 6233	. . . . . . . 8 ⊢ (𝑓 = (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) → (𝑓:𝑈⟶𝐽 ↔ (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})):𝑈⟶𝐽))
119		rneq 5552	. . . . . . . . 9 ⊢ (𝑓 = (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) → ran 𝑓 = ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})))
120	119	breq1d 4854	. . . . . . . 8 ⊢ (𝑓 = (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) → (ran 𝑓Ref𝑈 ↔ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅}))Ref𝑈))
121	119	eleq1d 2870	. . . . . . . 8 ⊢ (𝑓 = (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) → (ran 𝑓 ∈ (LocFin‘𝐽) ↔ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ (LocFin‘𝐽)))
122	118, 120, 121	3anbi123d 1553	. . . . . . 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 3487	. . . . . 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 395	. . . 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 1484	. . 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 2026	. 2 ⊢ ((𝜑 ∧ 𝑈 ≠ ∅) → ∃𝑓(𝑓:𝑈⟶𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
128	43, 127	pm2.61dane 3065	1 ⊢ (𝜑 → ∃𝑓(𝑓:𝑈⟶𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))

Syntax hints:   → wi 4   ↔ wb 197   ∧ wa 384   ∨ wo 865   ∧ w3a 1100   = wceq 1637  ∃wex 1859   ∈ wcel 2156   ≠ wne 2978  ∀wral 3096  ∃wrex 3097  {crab 3100  Vcvv 3391   ∖ cdif 3766   ∪ cun 3767   ∩ cin 3768   ⊆ wss 3769  ∅c0 4116  {csn 4370  ∪ cuni 4630   class class class wbr 4844   × cxp 5309  dom cdm 5311  ran crn 5312  Fun wfun 6091  ⟶wf 6093  ‘cfv 6097  Fincfn 8188  Topctop 20908  Refcref 21516  LocFinclocfin 21518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175  ax-reg 8732  ax-inf2 8781  ax-ac2 9566

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-iin 4715  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-se 5271  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-isom 6106  df-riota 6831  df-ov 6873  df-oprab 6874  df-mpt2 6875  df-om 7292  df-wrecs 7638  df-recs 7700  df-rdg 7738  df-1o 7792  df-oadd 7796  df-er 7975  df-en 8189  df-dom 8190  df-fin 8192  df-r1 8870  df-rank 8871  df-card 9044  df-ac 9218  df-top 20909  df-topon 20926  df-ref 21519  df-locfin 21521

This theorem is referenced by:  pcmplfinf  30252