Theorem locfinreflem 34298

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. The solution is constructed by building unions, so the same method can be used to prove a similar theorem about closed covers. (Contributed by Thierry Arnoux, 29-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
locfinreflem	⊢ (𝜑 → ∃𝑓((Fun 𝑓 ∧ dom 𝑓 ⊆ 𝑈 ∧ ran 𝑓 ⊆ 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 locfinreflem

Dummy variables 𝑔 𝑗 𝑘 𝑢 𝑣 𝑤 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		locfinref.4	. . . 4 ⊢ (𝜑 → 𝑉Ref𝑈)
2		locfinref.5	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ (LocFin‘𝐽))
3		reff 34297	. . . . 5 ⊢ (𝑉 ∈ (LocFin‘𝐽) → (𝑉Ref𝑈 ↔ (∪ 𝑈 ⊆ ∪ 𝑉 ∧ ∃𝑔(𝑔:𝑉⟶𝑈 ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)))))
4	2, 3	syl 18	. . . 4 ⊢ (𝜑 → (𝑉Ref𝑈 ↔ (∪ 𝑈 ⊆ ∪ 𝑉 ∧ ∃𝑔(𝑔:𝑉⟶𝑈 ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)))))
5	1, 4	mpbid 235	. . 3 ⊢ (𝜑 → (∪ 𝑈 ⊆ ∪ 𝑉 ∧ ∃𝑔(𝑔:𝑉⟶𝑈 ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣))))
6	5	simprd 501	. 2 ⊢ (𝜑 → ∃𝑔(𝑔:𝑉⟶𝑈 ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)))
7		funmpt 6578	. . . . . 6 ⊢ Fun (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))
8	7	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) → Fun (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})))
9		eqid 2765	. . . . . . 7 ⊢ (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) = (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))
10	9	dmmptss 6244	. . . . . 6 ⊢ dom (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ⊆ ran 𝑔
11		frn 6717	. . . . . . 7 ⊢ (𝑔:𝑉⟶𝑈 → ran 𝑔 ⊆ 𝑈)
12	11	ad2antlr 740	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) → ran 𝑔 ⊆ 𝑈)
13	10, 12	sstrid 3949	. . . . 5 ⊢ (((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) → dom (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ⊆ 𝑈)
14		locfintop 23733	. . . . . . . . . 10 ⊢ (𝑉 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top)
15	2, 14	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ Top)
16	15	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑢 ∈ ran 𝑔) → 𝐽 ∈ Top)
17		cnvimass 6086	. . . . . . . . . 10 ⊢ (◡𝑔 “ {𝑢}) ⊆ dom 𝑔
18		fdm 6719	. . . . . . . . . . 11 ⊢ (𝑔:𝑉⟶𝑈 → dom 𝑔 = 𝑉)
19	18	ad3antlr 744	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑢 ∈ ran 𝑔) → dom 𝑔 = 𝑉)
20	17, 19	sseqtrid 3980	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑢 ∈ ran 𝑔) → (◡𝑔 “ {𝑢}) ⊆ 𝑉)
21		locfinref.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ⊆ 𝐽)
22	21	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑢 ∈ ran 𝑔) → 𝑉 ⊆ 𝐽)
23	20, 22	sstrd 3948	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑢 ∈ ran 𝑔) → (◡𝑔 “ {𝑢}) ⊆ 𝐽)
24		uniopn 23108	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (◡𝑔 “ {𝑢}) ⊆ 𝐽) → ∪ (◡𝑔 “ {𝑢}) ∈ 𝐽)
25	16, 23, 24	syl2anc 596	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑢 ∈ ran 𝑔) → ∪ (◡𝑔 “ {𝑢}) ∈ 𝐽)
26	25	ralrimiva 3159	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) → ∀𝑢 ∈ ran 𝑔∪ (◡𝑔 “ {𝑢}) ∈ 𝐽)
27	9	rnmptss 7122	. . . . . 6 ⊢ (∀𝑢 ∈ ran 𝑔∪ (◡𝑔 “ {𝑢}) ∈ 𝐽 → ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ⊆ 𝐽)
28	26, 27	syl 18	. . . . 5 ⊢ (((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) → ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ⊆ 𝐽)
29		eqid 2765	. . . . . . . . . 10 ⊢ ∪ 𝑉 = ∪ 𝑉
30		eqid 2765	. . . . . . . . . 10 ⊢ ∪ 𝑈 = ∪ 𝑈
31	29, 30	refbas 23722	. . . . . . . . 9 ⊢ (𝑉Ref𝑈 → ∪ 𝑈 = ∪ 𝑉)
32	1, 31	syl 18	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑈 = ∪ 𝑉)
33	32	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) → ∪ 𝑈 = ∪ 𝑉)
34		nfv 1947	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑣(𝜑 ∧ 𝑔:𝑉⟶𝑈)
35		nfra1 3291	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑣∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)
36	34, 35	nfan 1932	. . . . . . . . . . . 12 ⊢ Ⅎ𝑣((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣))
37		nfre1 3292	. . . . . . . . . . . 12 ⊢ Ⅎ𝑣∃𝑣 ∈ 𝑉 𝑥 ∈ 𝑣
38	36, 37	nfan 1932	. . . . . . . . . . 11 ⊢ Ⅎ𝑣(((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ ∃𝑣 ∈ 𝑉 𝑥 ∈ 𝑣)
39		ffn 6709	. . . . . . . . . . . . . . 15 ⊢ (𝑔:𝑉⟶𝑈 → 𝑔 Fn 𝑉)
40	39	ad4antlr 746	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑣 ∈ 𝑉) ∧ 𝑥 ∈ 𝑣) → 𝑔 Fn 𝑉)
41		simplr 781	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑣 ∈ 𝑉) ∧ 𝑥 ∈ 𝑣) → 𝑣 ∈ 𝑉)
42		fnfvelrn 7079	. . . . . . . . . . . . . 14 ⊢ ((𝑔 Fn 𝑉 ∧ 𝑣 ∈ 𝑉) → (𝑔‘𝑣) ∈ ran 𝑔)
43	40, 41, 42	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑣 ∈ 𝑉) ∧ 𝑥 ∈ 𝑣) → (𝑔‘𝑣) ∈ ran 𝑔)
44		ssid 3960	. . . . . . . . . . . . . . 15 ⊢ 𝑣 ⊆ 𝑣
45	39	ad3antlr 744	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑣 ∈ 𝑉) → 𝑔 Fn 𝑉)
46		eqid 2765	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔‘𝑣) = (𝑔‘𝑣)
47		fniniseg 7059	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 Fn 𝑉 → (𝑣 ∈ (◡𝑔 “ {(𝑔‘𝑣)}) ↔ (𝑣 ∈ 𝑉 ∧ (𝑔‘𝑣) = (𝑔‘𝑣))))
48	47	biimpar 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔 Fn 𝑉 ∧ (𝑣 ∈ 𝑉 ∧ (𝑔‘𝑣) = (𝑔‘𝑣))) → 𝑣 ∈ (◡𝑔 “ {(𝑔‘𝑣)}))
49	46, 48	mpanr2 717	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 Fn 𝑉 ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ (◡𝑔 “ {(𝑔‘𝑣)}))
50	45, 49	sylancom 600	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ (◡𝑔 “ {(𝑔‘𝑣)}))
51		ssuni 4900	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 ⊆ 𝑣 ∧ 𝑣 ∈ (◡𝑔 “ {(𝑔‘𝑣)})) → 𝑣 ⊆ ∪ (◡𝑔 “ {(𝑔‘𝑣)}))
52	44, 50, 51	sylancr 599	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑣 ∈ 𝑉) → 𝑣 ⊆ ∪ (◡𝑔 “ {(𝑔‘𝑣)}))
53	52	sselda 3938	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑣 ∈ 𝑉) ∧ 𝑥 ∈ 𝑣) → 𝑥 ∈ ∪ (◡𝑔 “ {(𝑔‘𝑣)}))
54		sneq 4601	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = (𝑔‘𝑣) → {𝑢} = {(𝑔‘𝑣)})
55	54	imaeq2d 6064	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = (𝑔‘𝑣) → (◡𝑔 “ {𝑢}) = (◡𝑔 “ {(𝑔‘𝑣)}))
56	55	unieqd 4887	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (𝑔‘𝑣) → ∪ (◡𝑔 “ {𝑢}) = ∪ (◡𝑔 “ {(𝑔‘𝑣)}))
57	56	eleq2d 2851	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑔‘𝑣) → (𝑥 ∈ ∪ (◡𝑔 “ {𝑢}) ↔ 𝑥 ∈ ∪ (◡𝑔 “ {(𝑔‘𝑣)})))
58	57	rspcev 3583	. . . . . . . . . . . . 13 ⊢ (((𝑔‘𝑣) ∈ ran 𝑔 ∧ 𝑥 ∈ ∪ (◡𝑔 “ {(𝑔‘𝑣)})) → ∃𝑢 ∈ ran 𝑔 𝑥 ∈ ∪ (◡𝑔 “ {𝑢}))
59	43, 53, 58	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑣 ∈ 𝑉) ∧ 𝑥 ∈ 𝑣) → ∃𝑢 ∈ ran 𝑔 𝑥 ∈ ∪ (◡𝑔 “ {𝑢}))
60	59	adantllr 732	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ ∃𝑣 ∈ 𝑉 𝑥 ∈ 𝑣) ∧ 𝑣 ∈ 𝑉) ∧ 𝑥 ∈ 𝑣) → ∃𝑢 ∈ ran 𝑔 𝑥 ∈ ∪ (◡𝑔 “ {𝑢}))
61		simpr 490	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ ∃𝑣 ∈ 𝑉 𝑥 ∈ 𝑣) → ∃𝑣 ∈ 𝑉 𝑥 ∈ 𝑣)
62	38, 60, 61	r19.29af 3276	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ ∃𝑣 ∈ 𝑉 𝑥 ∈ 𝑣) → ∃𝑢 ∈ ran 𝑔 𝑥 ∈ ∪ (◡𝑔 “ {𝑢}))
63		nfv 1947	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑣 𝑢 ∈ ran 𝑔
64	36, 63	nfan 1932	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑣(((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑢 ∈ ran 𝑔)
65		nfv 1947	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑣 𝑥 ∈ ∪ (◡𝑔 “ {𝑢})
66	64, 65	nfan 1932	. . . . . . . . . . . 12 ⊢ Ⅎ𝑣((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 ∈ ∪ (◡𝑔 “ {𝑢}))
67	20	ad3antrrr 743	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 ∈ ∪ (◡𝑔 “ {𝑢})) ∧ 𝑣 ∈ (◡𝑔 “ {𝑢})) ∧ 𝑥 ∈ 𝑣) → (◡𝑔 “ {𝑢}) ⊆ 𝑉)
68		simplr 781	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 ∈ ∪ (◡𝑔 “ {𝑢})) ∧ 𝑣 ∈ (◡𝑔 “ {𝑢})) ∧ 𝑥 ∈ 𝑣) → 𝑣 ∈ (◡𝑔 “ {𝑢}))
69	67, 68	sseldd 3939	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 ∈ ∪ (◡𝑔 “ {𝑢})) ∧ 𝑣 ∈ (◡𝑔 “ {𝑢})) ∧ 𝑥 ∈ 𝑣) → 𝑣 ∈ 𝑉)
70		simpr 490	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 ∈ ∪ (◡𝑔 “ {𝑢})) ∧ 𝑣 ∈ (◡𝑔 “ {𝑢})) ∧ 𝑥 ∈ 𝑣) → 𝑥 ∈ 𝑣)
71		eluni2 4878	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ∪ (◡𝑔 “ {𝑢}) ↔ ∃𝑣 ∈ (◡𝑔 “ {𝑢})𝑥 ∈ 𝑣)
72	71	bilani 510	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 ∈ ∪ (◡𝑔 “ {𝑢})) → ∃𝑣 ∈ (◡𝑔 “ {𝑢})𝑥 ∈ 𝑣)
73	66, 69, 70, 72	reximd2a 3277	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 ∈ ∪ (◡𝑔 “ {𝑢})) → ∃𝑣 ∈ 𝑉 𝑥 ∈ 𝑣)
74	73	r19.29an 3171	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ ∃𝑢 ∈ ran 𝑔 𝑥 ∈ ∪ (◡𝑔 “ {𝑢})) → ∃𝑣 ∈ 𝑉 𝑥 ∈ 𝑣)
75	62, 74	impbida 813	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) → (∃𝑣 ∈ 𝑉 𝑥 ∈ 𝑣 ↔ ∃𝑢 ∈ ran 𝑔 𝑥 ∈ ∪ (◡𝑔 “ {𝑢})))
76		eluni2 4878	. . . . . . . . 9 ⊢ (𝑥 ∈ ∪ 𝑉 ↔ ∃𝑣 ∈ 𝑉 𝑥 ∈ 𝑣)
77		eliun 4962	. . . . . . . . 9 ⊢ (𝑥 ∈ ∪ 𝑢 ∈ ran 𝑔∪ (◡𝑔 “ {𝑢}) ↔ ∃𝑢 ∈ ran 𝑔 𝑥 ∈ ∪ (◡𝑔 “ {𝑢}))
78	75, 76, 77	3bitr4g 317	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) → (𝑥 ∈ ∪ 𝑉 ↔ 𝑥 ∈ ∪ 𝑢 ∈ ran 𝑔∪ (◡𝑔 “ {𝑢})))
79	78	eqrdv 2763	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) → ∪ 𝑉 = ∪ 𝑢 ∈ ran 𝑔∪ (◡𝑔 “ {𝑢}))
80		dfiun3g 5960	. . . . . . . 8 ⊢ (∀𝑢 ∈ ran 𝑔∪ (◡𝑔 “ {𝑢}) ∈ 𝐽 → ∪ 𝑢 ∈ ran 𝑔∪ (◡𝑔 “ {𝑢}) = ∪ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})))
81	26, 80	syl 18	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) → ∪ 𝑢 ∈ ran 𝑔∪ (◡𝑔 “ {𝑢}) = ∪ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})))
82	33, 79, 81	3eqtrd 2804	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) → ∪ 𝑈 = ∪ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})))
83	11	ad3antlr 744	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))) → ran 𝑔 ⊆ 𝑈)
84		vex 3461	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
85	9	elrnmpt 5950	. . . . . . . . . . 11 ⊢ (𝑤 ∈ V → (𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ↔ ∃𝑢 ∈ ran 𝑔 𝑤 = ∪ (◡𝑔 “ {𝑢})))
86	84, 85	mp1i 14	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) → (𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ↔ ∃𝑢 ∈ ran 𝑔 𝑤 = ∪ (◡𝑔 “ {𝑢})))
87	86	biimpa 482	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))) → ∃𝑢 ∈ ran 𝑔 𝑤 = ∪ (◡𝑔 “ {𝑢}))
88		ssrexv 4008	. . . . . . . . 9 ⊢ (ran 𝑔 ⊆ 𝑈 → (∃𝑢 ∈ ran 𝑔 𝑤 = ∪ (◡𝑔 “ {𝑢}) → ∃𝑢 ∈ 𝑈 𝑤 = ∪ (◡𝑔 “ {𝑢})))
89	83, 87, 88	sylc 66	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))) → ∃𝑢 ∈ 𝑈 𝑤 = ∪ (◡𝑔 “ {𝑢}))
90		nfv 1947	. . . . . . . . . 10 ⊢ Ⅎ𝑢((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣))
91		nfmpt1 5212	. . . . . . . . . . . 12 ⊢ Ⅎ𝑢(𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))
92	91	nfrn 5944	. . . . . . . . . . 11 ⊢ Ⅎ𝑢ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))
93	92	nfcri 2919	. . . . . . . . . 10 ⊢ Ⅎ𝑢 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))
94	90, 93	nfan 1932	. . . . . . . . 9 ⊢ Ⅎ𝑢(((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})))
95		simpr 490	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑤 = ∪ (◡𝑔 “ {𝑢})) → 𝑤 = ∪ (◡𝑔 “ {𝑢}))
96		nfv 1947	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑣 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))
97	36, 96	nfan 1932	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑣(((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})))
98		nfv 1947	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑣 𝑢 ∈ 𝑈
99	97, 98	nfan 1932	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑣((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))) ∧ 𝑢 ∈ 𝑈)
100		nfv 1947	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑣 𝑤 = ∪ (◡𝑔 “ {𝑢})
101	99, 100	nfan 1932	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑣(((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑤 = ∪ (◡𝑔 “ {𝑢}))
102		simp-5r 798	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑤 = ∪ (◡𝑔 “ {𝑢})) ∧ 𝑣 ∈ (◡𝑔 “ {𝑢})) → ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣))
103	39	ad5antlr 748	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑤 = ∪ (◡𝑔 “ {𝑢})) → 𝑔 Fn 𝑉)
104		fniniseg 7059	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 Fn 𝑉 → (𝑣 ∈ (◡𝑔 “ {𝑢}) ↔ (𝑣 ∈ 𝑉 ∧ (𝑔‘𝑣) = 𝑢)))
105	103, 104	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑤 = ∪ (◡𝑔 “ {𝑢})) → (𝑣 ∈ (◡𝑔 “ {𝑢}) ↔ (𝑣 ∈ 𝑉 ∧ (𝑔‘𝑣) = 𝑢)))
106	105	biimpa 482	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑤 = ∪ (◡𝑔 “ {𝑢})) ∧ 𝑣 ∈ (◡𝑔 “ {𝑢})) → (𝑣 ∈ 𝑉 ∧ (𝑔‘𝑣) = 𝑢))
107	106	simpld 500	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑤 = ∪ (◡𝑔 “ {𝑢})) ∧ 𝑣 ∈ (◡𝑔 “ {𝑢})) → 𝑣 ∈ 𝑉)
108		rspa 3256	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣) ∧ 𝑣 ∈ 𝑉) → 𝑣 ⊆ (𝑔‘𝑣))
109	102, 107, 108	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑤 = ∪ (◡𝑔 “ {𝑢})) ∧ 𝑣 ∈ (◡𝑔 “ {𝑢})) → 𝑣 ⊆ (𝑔‘𝑣))
110	106	simprd 501	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑤 = ∪ (◡𝑔 “ {𝑢})) ∧ 𝑣 ∈ (◡𝑔 “ {𝑢})) → (𝑔‘𝑣) = 𝑢)
111	109, 110	sseqtrd 3974	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑤 = ∪ (◡𝑔 “ {𝑢})) ∧ 𝑣 ∈ (◡𝑔 “ {𝑢})) → 𝑣 ⊆ 𝑢)
112	111	ex 418	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑤 = ∪ (◡𝑔 “ {𝑢})) → (𝑣 ∈ (◡𝑔 “ {𝑢}) → 𝑣 ⊆ 𝑢))
113	101, 112	ralrimi 3265	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑤 = ∪ (◡𝑔 “ {𝑢})) → ∀𝑣 ∈ (◡𝑔 “ {𝑢})𝑣 ⊆ 𝑢)
114		unissb 4908	. . . . . . . . . . . 12 ⊢ (∪ (◡𝑔 “ {𝑢}) ⊆ 𝑢 ↔ ∀𝑣 ∈ (◡𝑔 “ {𝑢})𝑣 ⊆ 𝑢)
115	113, 114	sylibr 237	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑤 = ∪ (◡𝑔 “ {𝑢})) → ∪ (◡𝑔 “ {𝑢}) ⊆ 𝑢)
116	95, 115	eqsstrd 3972	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))) ∧ 𝑢 ∈ 𝑈) ∧ 𝑤 = ∪ (◡𝑔 “ {𝑢})) → 𝑤 ⊆ 𝑢)
117	116	exp31 425	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))) → (𝑢 ∈ 𝑈 → (𝑤 = ∪ (◡𝑔 “ {𝑢}) → 𝑤 ⊆ 𝑢)))
118	94, 117	reximdai 3269	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))) → (∃𝑢 ∈ 𝑈 𝑤 = ∪ (◡𝑔 “ {𝑢}) → ∃𝑢 ∈ 𝑈 𝑤 ⊆ 𝑢))
119	89, 118	mpd 16	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))) → ∃𝑢 ∈ 𝑈 𝑤 ⊆ 𝑢)
120	119	ralrimiva 3159	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) → ∀𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))∃𝑢 ∈ 𝑈 𝑤 ⊆ 𝑢)
121		vex 3461	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
122	121	rnex 7913	. . . . . . . . 9 ⊢ ran 𝑔 ∈ V
123	122	mptex 7228	. . . . . . . 8 ⊢ (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ∈ V
124		rnexg 7905	. . . . . . . 8 ⊢ ((𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ∈ V → ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ∈ V)
125	123, 124	mp1i 14	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) → ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ∈ V)
126		eqid 2765	. . . . . . . 8 ⊢ ∪ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) = ∪ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))
127	126, 30	isref 23721	. . . . . . 7 ⊢ (ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ∈ V → (ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))Ref𝑈 ↔ (∪ 𝑈 = ∪ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ∧ ∀𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))∃𝑢 ∈ 𝑈 𝑤 ⊆ 𝑢)))
128	125, 127	syl 18	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) → (ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))Ref𝑈 ↔ (∪ 𝑈 = ∪ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ∧ ∀𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))∃𝑢 ∈ 𝑈 𝑤 ⊆ 𝑢)))
129	82, 120, 128	mpbir2and 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) → ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))Ref𝑈)
130	15	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) → 𝐽 ∈ Top)
131		locfinref.2	. . . . . . . 8 ⊢ (𝜑 → 𝑋 = ∪ 𝑈)
132	131	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) → 𝑋 = ∪ 𝑈)
133	132, 82	eqtrd 2800	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) → 𝑋 = ∪ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})))
134		nfv 1947	. . . . . . . . 9 ⊢ Ⅎ𝑣 𝑥 ∈ 𝑋
135	36, 134	nfan 1932	. . . . . . . 8 ⊢ Ⅎ𝑣(((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋)
136		simplr 781	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ (𝑥 ∈ 𝑣 ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin)) → 𝑣 ∈ 𝐽)
137		ffun 6712	. . . . . . . . . . . . . 14 ⊢ (𝑔:𝑉⟶𝑈 → Fun 𝑔)
138	137	ad6antlr 750	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin) → Fun 𝑔)
139		imafi 9282	. . . . . . . . . . . . 13 ⊢ ((Fun 𝑔 ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin) → (𝑔 “ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅}) ∈ Fin)
140	138, 139	sylancom 600	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin) → (𝑔 “ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅}) ∈ Fin)
141		simp3 1156	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin) ∧ 𝑘 ∈ ran 𝑔 ∧ 𝑤 = ((𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))‘𝑘)) → 𝑤 = ((𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))‘𝑘))
142		sneq 4601	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑢 = 𝑘 → {𝑢} = {𝑘})
143	142	imaeq2d 6064	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑢 = 𝑘 → (◡𝑔 “ {𝑢}) = (◡𝑔 “ {𝑘}))
144	143	unieqd 4887	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 = 𝑘 → ∪ (◡𝑔 “ {𝑢}) = ∪ (◡𝑔 “ {𝑘}))
145	121	cnvex 7928	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ◡𝑔 ∈ V
146		imaexg 7916	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (◡𝑔 ∈ V → (◡𝑔 “ {𝑘}) ∈ V)
147	145, 146	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (◡𝑔 “ {𝑘}) ∈ V
148	147	uniex 7749	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∪ (◡𝑔 “ {𝑘}) ∈ V
149	144, 9, 148	fvmpt 6993	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ ran 𝑔 → ((𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))‘𝑘) = ∪ (◡𝑔 “ {𝑘}))
150	149	3ad2ant2 1152	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin) ∧ 𝑘 ∈ ran 𝑔 ∧ 𝑤 = ((𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))‘𝑘)) → ((𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))‘𝑘) = ∪ (◡𝑔 “ {𝑘}))
151	141, 150	eqtrd 2800	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin) ∧ 𝑘 ∈ ran 𝑔 ∧ 𝑤 = ((𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))‘𝑘)) → 𝑤 = ∪ (◡𝑔 “ {𝑘}))
152	151	ineq1d 4172	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin) ∧ 𝑘 ∈ ran 𝑔 ∧ 𝑤 = ((𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))‘𝑘)) → (𝑤 ∩ 𝑣) = (∪ (◡𝑔 “ {𝑘}) ∩ 𝑣))
153	152	neeq1d 3019	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin) ∧ 𝑘 ∈ ran 𝑔 ∧ 𝑤 = ((𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))‘𝑘)) → ((𝑤 ∩ 𝑣) ≠ ∅ ↔ (∪ (◡𝑔 “ {𝑘}) ∩ 𝑣) ≠ ∅))
154	122	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin) → ran 𝑔 ∈ V)
155		imaexg 7916	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (◡𝑔 ∈ V → (◡𝑔 “ {𝑢}) ∈ V)
156	145, 155	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ (◡𝑔 “ {𝑢}) ∈ V
157	156	uniex 7749	. . . . . . . . . . . . . . . . . . 19 ⊢ ∪ (◡𝑔 “ {𝑢}) ∈ V
158	157, 9	fnmpti 6682	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) Fn ran 𝑔
159		dffn4 6802	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) Fn ran 𝑔 ↔ (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})):ran 𝑔–onto→ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})))
160	158, 159	mpbi 233	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})):ran 𝑔–onto→ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))
161	160	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin) → (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})):ran 𝑔–onto→ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})))
162	153, 154, 161	rabfodom 32926	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin) → {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ∣ (𝑤 ∩ 𝑣) ≠ ∅} ≼ {𝑘 ∈ ran 𝑔 ∣ (∪ (◡𝑔 “ {𝑘}) ∩ 𝑣) ≠ ∅})
163		sneq 4601	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑢 → {𝑘} = {𝑢})
164	163	imaeq2d 6064	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑢 → (◡𝑔 “ {𝑘}) = (◡𝑔 “ {𝑢}))
165	164	unieqd 4887	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑢 → ∪ (◡𝑔 “ {𝑘}) = ∪ (◡𝑔 “ {𝑢}))
166	165	ineq1d 4172	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑢 → (∪ (◡𝑔 “ {𝑘}) ∩ 𝑣) = (∪ (◡𝑔 “ {𝑢}) ∩ 𝑣))
167	166	neeq1d 3019	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑢 → ((∪ (◡𝑔 “ {𝑘}) ∩ 𝑣) ≠ ∅ ↔ (∪ (◡𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅))
168	167	cbvrabv 3428	. . . . . . . . . . . . . . 15 ⊢ {𝑘 ∈ ran 𝑔 ∣ (∪ (◡𝑔 “ {𝑘}) ∩ 𝑣) ≠ ∅} = {𝑢 ∈ ran 𝑔 ∣ (∪ (◡𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅}
169	162, 168	breqtrdi 5154	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin) → {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ∣ (𝑤 ∩ 𝑣) ≠ ∅} ≼ {𝑢 ∈ ran 𝑔 ∣ (∪ (◡𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅})
170	122	rabex 5311	. . . . . . . . . . . . . . 15 ⊢ {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} (𝑔‘𝑘) = 𝑢} ∈ V
171		nfv 1947	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑗(((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣)
172		nfrab1 3438	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑗{𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅}
173	172	nfel1 2943	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑗{𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin
174	171, 173	nfan 1932	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑗((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin)
175		nfv 1947	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑗 𝑢 ∈ ran 𝑔
176	174, 175	nfan 1932	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑗(((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔)
177		nfv 1947	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑗(∪ (◡𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅
178	176, 177	nfan 1932	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑗((((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ (∪ (◡𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅)
179		nfv 1947	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑗(𝑔‘𝑘) = 𝑢
180	172, 179	nfrexw 3315	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑗∃𝑘 ∈ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} (𝑔‘𝑘) = 𝑢
181	39	ad5antlr 748	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → 𝑔 Fn 𝑉)
182	181	ad5antr 747	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ (∪ (◡𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) ∧ 𝑗 ∈ (◡𝑔 “ {𝑢})) ∧ (𝑗 ∩ 𝑣) ≠ ∅) → 𝑔 Fn 𝑉)
183		simplr 781	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ (∪ (◡𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) ∧ 𝑗 ∈ (◡𝑔 “ {𝑢})) ∧ (𝑗 ∩ 𝑣) ≠ ∅) → 𝑗 ∈ (◡𝑔 “ {𝑢}))
184		fniniseg 7059	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑔 Fn 𝑉 → (𝑗 ∈ (◡𝑔 “ {𝑢}) ↔ (𝑗 ∈ 𝑉 ∧ (𝑔‘𝑗) = 𝑢)))
185	184	biimpa 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑔 Fn 𝑉 ∧ 𝑗 ∈ (◡𝑔 “ {𝑢})) → (𝑗 ∈ 𝑉 ∧ (𝑔‘𝑗) = 𝑢))
186	182, 183, 185	syl2anc 596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ (∪ (◡𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) ∧ 𝑗 ∈ (◡𝑔 “ {𝑢})) ∧ (𝑗 ∩ 𝑣) ≠ ∅) → (𝑗 ∈ 𝑉 ∧ (𝑔‘𝑗) = 𝑢))
187	186	simpld 500	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ (∪ (◡𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) ∧ 𝑗 ∈ (◡𝑔 “ {𝑢})) ∧ (𝑗 ∩ 𝑣) ≠ ∅) → 𝑗 ∈ 𝑉)
188		simpr 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ (∪ (◡𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) ∧ 𝑗 ∈ (◡𝑔 “ {𝑢})) ∧ (𝑗 ∩ 𝑣) ≠ ∅) → (𝑗 ∩ 𝑣) ≠ ∅)
189		rabid 3439	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ↔ (𝑗 ∈ 𝑉 ∧ (𝑗 ∩ 𝑣) ≠ ∅))
190	187, 188, 189	sylanbrc 595	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ (∪ (◡𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) ∧ 𝑗 ∈ (◡𝑔 “ {𝑢})) ∧ (𝑗 ∩ 𝑣) ≠ ∅) → 𝑗 ∈ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅})
191	186	simprd 501	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ (∪ (◡𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) ∧ 𝑗 ∈ (◡𝑔 “ {𝑢})) ∧ (𝑗 ∩ 𝑣) ≠ ∅) → (𝑔‘𝑗) = 𝑢)
192		fveqeq2 6894	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑗 → ((𝑔‘𝑘) = 𝑢 ↔ (𝑔‘𝑗) = 𝑢))
193	192	rspcev 3583	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∧ (𝑔‘𝑗) = 𝑢) → ∃𝑘 ∈ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} (𝑔‘𝑘) = 𝑢)
194	190, 191, 193	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ (∪ (◡𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) ∧ 𝑗 ∈ (◡𝑔 “ {𝑢})) ∧ (𝑗 ∩ 𝑣) ≠ ∅) → ∃𝑘 ∈ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} (𝑔‘𝑘) = 𝑢)
195		uniinn0 32972	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∪ (◡𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅ ↔ ∃𝑗 ∈ (◡𝑔 “ {𝑢})(𝑗 ∩ 𝑣) ≠ ∅)
196	195	bilani 510	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ (∪ (◡𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) → ∃𝑗 ∈ (◡𝑔 “ {𝑢})(𝑗 ∩ 𝑣) ≠ ∅)
197	178, 180, 194, 196	r19.29af2 3275	. . . . . . . . . . . . . . . . 17 ⊢ (((((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ (∪ (◡𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) → ∃𝑘 ∈ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} (𝑔‘𝑘) = 𝑢)
198	197	ex 418	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) → ((∪ (◡𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅ → ∃𝑘 ∈ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} (𝑔‘𝑘) = 𝑢))
199	198	ss2rabdv 4030	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin) → {𝑢 ∈ ran 𝑔 ∣ (∪ (◡𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅} ⊆ {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} (𝑔‘𝑘) = 𝑢})
200		ssdomg 9003	. . . . . . . . . . . . . . 15 ⊢ ({𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} (𝑔‘𝑘) = 𝑢} ∈ V → ({𝑢 ∈ ran 𝑔 ∣ (∪ (◡𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅} ⊆ {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} (𝑔‘𝑘) = 𝑢} → {𝑢 ∈ ran 𝑔 ∣ (∪ (◡𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅} ≼ {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} (𝑔‘𝑘) = 𝑢}))
201	170, 199, 200	mpsyl 69	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin) → {𝑢 ∈ ran 𝑔 ∣ (∪ (◡𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅} ≼ {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} (𝑔‘𝑘) = 𝑢})
202		domtr 9010	. . . . . . . . . . . . . 14 ⊢ (({𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ∣ (𝑤 ∩ 𝑣) ≠ ∅} ≼ {𝑢 ∈ ran 𝑔 ∣ (∪ (◡𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅} ∧ {𝑢 ∈ ran 𝑔 ∣ (∪ (◡𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅} ≼ {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} (𝑔‘𝑘) = 𝑢}) → {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ∣ (𝑤 ∩ 𝑣) ≠ ∅} ≼ {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} (𝑔‘𝑘) = 𝑢})
203	169, 201, 202	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin) → {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ∣ (𝑤 ∩ 𝑣) ≠ ∅} ≼ {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} (𝑔‘𝑘) = 𝑢})
204	181	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin) → 𝑔 Fn 𝑉)
205		dffn3 6722	. . . . . . . . . . . . . . 15 ⊢ (𝑔 Fn 𝑉 ↔ 𝑔:𝑉⟶ran 𝑔)
206	205	biimpi 219	. . . . . . . . . . . . . 14 ⊢ (𝑔 Fn 𝑉 → 𝑔:𝑉⟶ran 𝑔)
207		ssrab2 4035	. . . . . . . . . . . . . . 15 ⊢ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ⊆ 𝑉
208		fimarab 6959	. . . . . . . . . . . . . . 15 ⊢ ((𝑔:𝑉⟶ran 𝑔 ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ⊆ 𝑉) → (𝑔 “ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅}) = {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} (𝑔‘𝑘) = 𝑢})
209	207, 208	mpan2 704	. . . . . . . . . . . . . 14 ⊢ (𝑔:𝑉⟶ran 𝑔 → (𝑔 “ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅}) = {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} (𝑔‘𝑘) = 𝑢})
210	204, 206, 209	3syl 19	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin) → (𝑔 “ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅}) = {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} (𝑔‘𝑘) = 𝑢})
211	203, 210	breqtrrd 5141	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin) → {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ∣ (𝑤 ∩ 𝑣) ≠ ∅} ≼ (𝑔 “ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅}))
212		domfi 9180	. . . . . . . . . . . 12 ⊢ (((𝑔 “ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅}) ∈ Fin ∧ {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ∣ (𝑤 ∩ 𝑣) ≠ ∅} ≼ (𝑔 “ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅})) → {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ∣ (𝑤 ∩ 𝑣) ≠ ∅} ∈ Fin)
213	140, 211, 212	syl2anc 596	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin) → {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ∣ (𝑤 ∩ 𝑣) ≠ ∅} ∈ Fin)
214	213	ex 418	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → ({𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin → {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ∣ (𝑤 ∩ 𝑣) ≠ ∅} ∈ Fin))
215	214	imdistanda 582	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) → ((𝑥 ∈ 𝑣 ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin) → (𝑥 ∈ 𝑣 ∧ {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ∣ (𝑤 ∩ 𝑣) ≠ ∅} ∈ Fin)))
216	215	imp 412	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑣 ∈ 𝐽) ∧ (𝑥 ∈ 𝑣 ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin)) → (𝑥 ∈ 𝑣 ∧ {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ∣ (𝑤 ∩ 𝑣) ≠ ∅} ∈ Fin))
217		simplll 787	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) → 𝜑)
218		locfinref.x	. . . . . . . . . . . . 13 ⊢ 𝑋 = ∪ 𝐽
219	218, 29	islocfin 23729	. . . . . . . . . . . 12 ⊢ (𝑉 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑉 ∧ ∀𝑥 ∈ 𝑋 ∃𝑣 ∈ 𝐽 (𝑥 ∈ 𝑣 ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin)))
220	2, 219	sylib 221	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑉 ∧ ∀𝑥 ∈ 𝑋 ∃𝑣 ∈ 𝐽 (𝑥 ∈ 𝑣 ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin)))
221	220	simp3d 1162	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∃𝑣 ∈ 𝐽 (𝑥 ∈ 𝑣 ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin))
222	221	r19.21bi 3259	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∃𝑣 ∈ 𝐽 (𝑥 ∈ 𝑣 ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin))
223	217, 222	sylancom 600	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) → ∃𝑣 ∈ 𝐽 (𝑥 ∈ 𝑣 ∧ {𝑗 ∈ 𝑉 ∣ (𝑗 ∩ 𝑣) ≠ ∅} ∈ Fin))
224	135, 136, 216, 223	reximd2a 3277	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) ∧ 𝑥 ∈ 𝑋) → ∃𝑣 ∈ 𝐽 (𝑥 ∈ 𝑣 ∧ {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ∣ (𝑤 ∩ 𝑣) ≠ ∅} ∈ Fin))
225	224	ralrimiva 3159	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) → ∀𝑥 ∈ 𝑋 ∃𝑣 ∈ 𝐽 (𝑥 ∈ 𝑣 ∧ {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ∣ (𝑤 ∩ 𝑣) ≠ ∅} ∈ Fin))
226	218, 126	islocfin 23729	. . . . . 6 ⊢ (ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = ∪ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ∧ ∀𝑥 ∈ 𝑋 ∃𝑣 ∈ 𝐽 (𝑥 ∈ 𝑣 ∧ {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ∣ (𝑤 ∩ 𝑣) ≠ ∅} ∈ Fin)))
227	130, 133, 225, 226	syl3anbrc 1362	. . . . 5 ⊢ (((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) → ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ∈ (LocFin‘𝐽))
228		funeq 6560	. . . . . . . 8 ⊢ (𝑓 = (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) → (Fun 𝑓 ↔ Fun (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))))
229		dmeq 5895	. . . . . . . . 9 ⊢ (𝑓 = (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) → dom 𝑓 = dom (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})))
230	229	sseq1d 3969	. . . . . . . 8 ⊢ (𝑓 = (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) → (dom 𝑓 ⊆ 𝑈 ↔ dom (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ⊆ 𝑈))
231		rneq 5928	. . . . . . . . 9 ⊢ (𝑓 = (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) → ran 𝑓 = ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})))
232	231	sseq1d 3969	. . . . . . . 8 ⊢ (𝑓 = (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) → (ran 𝑓 ⊆ 𝐽 ↔ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ⊆ 𝐽))
233	228, 230, 232	3anbi123d 1464	. . . . . . 7 ⊢ (𝑓 = (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) → ((Fun 𝑓 ∧ dom 𝑓 ⊆ 𝑈 ∧ ran 𝑓 ⊆ 𝐽) ↔ (Fun (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ∧ dom (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ⊆ 𝑈 ∧ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ⊆ 𝐽)))
234	231	breq1d 5121	. . . . . . . 8 ⊢ (𝑓 = (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) → (ran 𝑓Ref𝑈 ↔ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))Ref𝑈))
235	231	eleq1d 2850	. . . . . . . 8 ⊢ (𝑓 = (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) → (ran 𝑓 ∈ (LocFin‘𝐽) ↔ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ∈ (LocFin‘𝐽)))
236	234, 235	anbi12d 644	. . . . . . 7 ⊢ (𝑓 = (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) → ((ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)) ↔ (ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))Ref𝑈 ∧ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ∈ (LocFin‘𝐽))))
237	233, 236	anbi12d 644	. . . . . 6 ⊢ (𝑓 = (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) → (((Fun 𝑓 ∧ dom 𝑓 ⊆ 𝑈 ∧ ran 𝑓 ⊆ 𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))) ↔ ((Fun (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ∧ dom (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ⊆ 𝑈 ∧ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ⊆ 𝐽) ∧ (ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))Ref𝑈 ∧ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ∈ (LocFin‘𝐽)))))
238	123, 237	spcev 3567	. . . . 5 ⊢ (((Fun (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ∧ dom (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ⊆ 𝑈 ∧ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ⊆ 𝐽) ∧ (ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢}))Ref𝑈 ∧ ran (𝑢 ∈ ran 𝑔 ↦ ∪ (◡𝑔 “ {𝑢})) ∈ (LocFin‘𝐽))) → ∃𝑓((Fun 𝑓 ∧ dom 𝑓 ⊆ 𝑈 ∧ ran 𝑓 ⊆ 𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))))
239	8, 13, 28, 129, 227, 238	syl32anc 1405	. . . 4 ⊢ (((𝜑 ∧ 𝑔:𝑉⟶𝑈) ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) → ∃𝑓((Fun 𝑓 ∧ dom 𝑓 ⊆ 𝑈 ∧ ran 𝑓 ⊆ 𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))))
240	239	expl 463	. . 3 ⊢ (𝜑 → ((𝑔:𝑉⟶𝑈 ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) → ∃𝑓((Fun 𝑓 ∧ dom 𝑓 ⊆ 𝑈 ∧ ran 𝑓 ⊆ 𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))))
241	240	exlimdv 1966	. 2 ⊢ (𝜑 → (∃𝑔(𝑔:𝑉⟶𝑈 ∧ ∀𝑣 ∈ 𝑉 𝑣 ⊆ (𝑔‘𝑣)) → ∃𝑓((Fun 𝑓 ∧ dom 𝑓 ⊆ 𝑈 ∧ ran 𝑓 ⊆ 𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))))
242	6, 241	mpd 16	1 ⊢ (𝜑 → ∃𝑓((Fun 𝑓 ∧ dom 𝑓 ⊆ 𝑈 ∧ ran 𝑓 ⊆ 𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  ∃wrex 3091  {crab 3418  Vcvv 3457   ∩ cin 3905   ⊆ wss 3906  ∅c0 4286  {csn 4591  ∪ cuni 4874  ∪ ciun 4958   class class class wbr 5111   ↦ cmpt 5194  ^◡ccnv 5662  dom cdm 5663  ran crn 5664   “ cima 5666  Fun wfun 6534   Fn wfn 6535  ⟶wf 6536  –onto→wfo 6538  ‘cfv 6540   ≼ cdom 8947  Fincfn 8949  Topctop 23104  Refcref 23714  LocFinclocfin 23716

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-reg 9561  ax-inf2 9617  ax-ac2 10462

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7376  df-ov 7422  df-om 7869  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-en 8950  df-dom 8951  df-fin 8953  df-r1 9743  df-rank 9744  df-scott 9865  df-card 9941  df-ac 10116  df-top 23105  df-ref 23717  df-locfin 23719

This theorem is used by:  locfinref  34299