Theorem cmpcref 34185

Description: Equivalent definition of compact space in terms of open cover refinements. Compact spaces are topologies with finite open cover refinements. (Contributed by Thierry Arnoux, 7-Jan-2020.)

Assertion

Ref	Expression
cmpcref	⊢ Comp = CovHasRefFin

Proof of Theorem cmpcref

Dummy variables 𝑓 𝑗 𝑢 𝑣 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 780	. . . . . . . . . . . . . . 15 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → 𝑥 ∈ (𝒫 𝑦 ∩ Fin))
2		elin 3927	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝒫 𝑦 ∩ Fin) ↔ (𝑥 ∈ 𝒫 𝑦 ∧ 𝑥 ∈ Fin))
3	1, 2	sylib 221	. . . . . . . . . . . . . 14 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → (𝑥 ∈ 𝒫 𝑦 ∧ 𝑥 ∈ Fin))
4	3	simpld 499	. . . . . . . . . . . . 13 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → 𝑥 ∈ 𝒫 𝑦)
5		elpwi 4572	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝒫 𝑦 → 𝑥 ⊆ 𝑦)
6	4, 5	syl 18	. . . . . . . . . . . 12 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → 𝑥 ⊆ 𝑦)
7		elpwi 4572	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝒫 𝑗 → 𝑦 ⊆ 𝑗)
8	7	ad4antlr 745	. . . . . . . . . . . 12 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → 𝑦 ⊆ 𝑗)
9	6, 8	sstrd 3953	. . . . . . . . . . 11 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → 𝑥 ⊆ 𝑗)
10		velpw 4570	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 𝑗 ↔ 𝑥 ⊆ 𝑗)
11	9, 10	sylibr 237	. . . . . . . . . 10 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → 𝑥 ∈ 𝒫 𝑗)
12	3	simprd 500	. . . . . . . . . 10 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → 𝑥 ∈ Fin)
13	11, 12	elind 4159	. . . . . . . . 9 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → 𝑥 ∈ (𝒫 𝑗 ∩ Fin))
14		simpr 489	. . . . . . . . . . 11 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → ∪ 𝑗 = ∪ 𝑥)
15		simpllr 787	. . . . . . . . . . 11 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → ∪ 𝑗 = ∪ 𝑦)
16	14, 15	eqtr3d 2806	. . . . . . . . . 10 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → ∪ 𝑥 = ∪ 𝑦)
17		eqid 2769	. . . . . . . . . . 11 ⊢ ∪ 𝑥 = ∪ 𝑥
18		eqid 2769	. . . . . . . . . . 11 ⊢ ∪ 𝑦 = ∪ 𝑦
19	17, 18	ssref 23638	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝒫 𝑗 ∧ 𝑥 ⊆ 𝑦 ∧ ∪ 𝑥 = ∪ 𝑦) → 𝑥Ref𝑦)
20	11, 6, 16, 19	syl3anc 1396	. . . . . . . . 9 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → 𝑥Ref𝑦)
21		breq1 5114	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑧Ref𝑦 ↔ 𝑥Ref𝑦))
22	21	rspcev 3588	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝒫 𝑗 ∩ Fin) ∧ 𝑥Ref𝑦) → ∃𝑧 ∈ (𝒫 𝑗 ∩ Fin)𝑧Ref𝑦)
23	13, 20, 22	syl2anc 595	. . . . . . . 8 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → ∃𝑧 ∈ (𝒫 𝑗 ∩ Fin)𝑧Ref𝑦)
24	23	r19.29an 3175	. . . . . . 7 ⊢ ((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ ∃𝑥 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑗 = ∪ 𝑥) → ∃𝑧 ∈ (𝒫 𝑗 ∩ Fin)𝑧Ref𝑦)
25		simplr 780	. . . . . . . . . 10 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) → 𝑧 ∈ (𝒫 𝑗 ∩ Fin))
26		vex 3465	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
27		eqid 2769	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑧 = ∪ 𝑧
28	27, 18	isref 23635	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ V → (𝑧Ref𝑦 ↔ (∪ 𝑦 = ∪ 𝑧 ∧ ∀𝑢 ∈ 𝑧 ∃𝑣 ∈ 𝑦 𝑢 ⊆ 𝑣)))
29	26, 28	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝑧Ref𝑦 ↔ (∪ 𝑦 = ∪ 𝑧 ∧ ∀𝑢 ∈ 𝑧 ∃𝑣 ∈ 𝑦 𝑢 ⊆ 𝑣))
30	29	simprbi 502	. . . . . . . . . . 11 ⊢ (𝑧Ref𝑦 → ∀𝑢 ∈ 𝑧 ∃𝑣 ∈ 𝑦 𝑢 ⊆ 𝑣)
31	30	adantl 486	. . . . . . . . . 10 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) → ∀𝑢 ∈ 𝑧 ∃𝑣 ∈ 𝑦 𝑢 ⊆ 𝑣)
32		sseq2 3969	. . . . . . . . . . 11 ⊢ (𝑣 = (𝑓‘𝑢) → (𝑢 ⊆ 𝑣 ↔ 𝑢 ⊆ (𝑓‘𝑢)))
33	32	ac6sg 10472	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝒫 𝑗 ∩ Fin) → (∀𝑢 ∈ 𝑧 ∃𝑣 ∈ 𝑦 𝑢 ⊆ 𝑣 → ∃𝑓(𝑓:𝑧⟶𝑦 ∧ ∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢))))
34	25, 31, 33	sylc 66	. . . . . . . . 9 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) → ∃𝑓(𝑓:𝑧⟶𝑦 ∧ ∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢)))
35		simplr 780	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) ∧ 𝑓:𝑧⟶𝑦) ∧ ∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢)) → 𝑓:𝑧⟶𝑦)
36	35	frnd 6715	. . . . . . . . . . . . . 14 ⊢ (((((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) ∧ 𝑓:𝑧⟶𝑦) ∧ ∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢)) → ran 𝑓 ⊆ 𝑦)
37		vex 3465	. . . . . . . . . . . . . . . 16 ⊢ 𝑓 ∈ V
38	37	rnex 7907	. . . . . . . . . . . . . . 15 ⊢ ran 𝑓 ∈ V
39	38	elpw 4569	. . . . . . . . . . . . . 14 ⊢ (ran 𝑓 ∈ 𝒫 𝑦 ↔ ran 𝑓 ⊆ 𝑦)
40	36, 39	sylibr 237	. . . . . . . . . . . . 13 ⊢ (((((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) ∧ 𝑓:𝑧⟶𝑦) ∧ ∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢)) → ran 𝑓 ∈ 𝒫 𝑦)
41	35	ffnd 6707	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) ∧ 𝑓:𝑧⟶𝑦) ∧ ∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢)) → 𝑓 Fn 𝑧)
42		elin 3927	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ (𝒫 𝑗 ∩ Fin) ↔ (𝑧 ∈ 𝒫 𝑗 ∧ 𝑧 ∈ Fin))
43	42	simprbi 502	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝒫 𝑗 ∩ Fin) → 𝑧 ∈ Fin)
44	43	ad4antlr 745	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) ∧ 𝑓:𝑧⟶𝑦) ∧ ∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢)) → 𝑧 ∈ Fin)
45		fnfi 9162	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 Fn 𝑧 ∧ 𝑧 ∈ Fin) → 𝑓 ∈ Fin)
46	41, 44, 45	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ (((((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) ∧ 𝑓:𝑧⟶𝑦) ∧ ∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢)) → 𝑓 ∈ Fin)
47		rnfi 9297	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ Fin → ran 𝑓 ∈ Fin)
48	46, 47	syl 18	. . . . . . . . . . . . 13 ⊢ (((((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) ∧ 𝑓:𝑧⟶𝑦) ∧ ∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢)) → ran 𝑓 ∈ Fin)
49	40, 48	elind 4159	. . . . . . . . . . . 12 ⊢ (((((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) ∧ 𝑓:𝑧⟶𝑦) ∧ ∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢)) → ran 𝑓 ∈ (𝒫 𝑦 ∩ Fin))
50		simp-5r 797	. . . . . . . . . . . . 13 ⊢ (((((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) ∧ 𝑓:𝑧⟶𝑦) ∧ ∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢)) → ∪ 𝑗 = ∪ 𝑦)
51	27, 18	refbas 23636	. . . . . . . . . . . . . . . 16 ⊢ (𝑧Ref𝑦 → ∪ 𝑦 = ∪ 𝑧)
52	51	ad3antlr 743	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) ∧ 𝑓:𝑧⟶𝑦) ∧ ∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢)) → ∪ 𝑦 = ∪ 𝑧)
53		nfv 1941	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑢(((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) ∧ 𝑓:𝑧⟶𝑦)
54		nfra1 3295	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑢∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢)
55	53, 54	nfan 1926	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑢((((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) ∧ 𝑓:𝑧⟶𝑦) ∧ ∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢))
56		rspa 3260	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢) ∧ 𝑢 ∈ 𝑧) → 𝑢 ⊆ (𝑓‘𝑢))
57	56	adantll 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) ∧ 𝑓:𝑧⟶𝑦) ∧ ∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢)) ∧ 𝑢 ∈ 𝑧) → 𝑢 ⊆ (𝑓‘𝑢))
58	57	sseld 3942	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) ∧ 𝑓:𝑧⟶𝑦) ∧ ∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢)) ∧ 𝑢 ∈ 𝑧) → (𝑥 ∈ 𝑢 → 𝑥 ∈ (𝑓‘𝑢)))
59	58	ex 417	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) ∧ 𝑓:𝑧⟶𝑦) ∧ ∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢)) → (𝑢 ∈ 𝑧 → (𝑥 ∈ 𝑢 → 𝑥 ∈ (𝑓‘𝑢))))
60	55, 59	reximdai 3273	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) ∧ 𝑓:𝑧⟶𝑦) ∧ ∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢)) → (∃𝑢 ∈ 𝑧 𝑥 ∈ 𝑢 → ∃𝑢 ∈ 𝑧 𝑥 ∈ (𝑓‘𝑢)))
61		eluni2 4878	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ∪ 𝑧 ↔ ∃𝑢 ∈ 𝑧 𝑥 ∈ 𝑢)
62	61	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) ∧ 𝑓:𝑧⟶𝑦) ∧ ∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢)) → (𝑥 ∈ ∪ 𝑧 ↔ ∃𝑢 ∈ 𝑧 𝑥 ∈ 𝑢))
63		fnunirn 7252	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 Fn 𝑧 → (𝑥 ∈ ∪ ran 𝑓 ↔ ∃𝑢 ∈ 𝑧 𝑥 ∈ (𝑓‘𝑢)))
64	41, 63	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) ∧ 𝑓:𝑧⟶𝑦) ∧ ∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢)) → (𝑥 ∈ ∪ ran 𝑓 ↔ ∃𝑢 ∈ 𝑧 𝑥 ∈ (𝑓‘𝑢)))
65	60, 62, 64	3imtr4d 297	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) ∧ 𝑓:𝑧⟶𝑦) ∧ ∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢)) → (𝑥 ∈ ∪ 𝑧 → 𝑥 ∈ ∪ ran 𝑓))
66	65	ssrdv 3949	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) ∧ 𝑓:𝑧⟶𝑦) ∧ ∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢)) → ∪ 𝑧 ⊆ ∪ ran 𝑓)
67	52, 66	eqsstrd 3977	. . . . . . . . . . . . . 14 ⊢ (((((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) ∧ 𝑓:𝑧⟶𝑦) ∧ ∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢)) → ∪ 𝑦 ⊆ ∪ ran 𝑓)
68	36	unissd 4884	. . . . . . . . . . . . . 14 ⊢ (((((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) ∧ 𝑓:𝑧⟶𝑦) ∧ ∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢)) → ∪ ran 𝑓 ⊆ ∪ 𝑦)
69	67, 68	eqssd 3960	. . . . . . . . . . . . 13 ⊢ (((((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) ∧ 𝑓:𝑧⟶𝑦) ∧ ∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢)) → ∪ 𝑦 = ∪ ran 𝑓)
70	50, 69	eqtrd 2804	. . . . . . . . . . . 12 ⊢ (((((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) ∧ 𝑓:𝑧⟶𝑦) ∧ ∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢)) → ∪ 𝑗 = ∪ ran 𝑓)
71		unieq 4885	. . . . . . . . . . . . 13 ⊢ (𝑥 = ran 𝑓 → ∪ 𝑥 = ∪ ran 𝑓)
72	71	rspceeqv 3611	. . . . . . . . . . . 12 ⊢ ((ran 𝑓 ∈ (𝒫 𝑦 ∩ Fin) ∧ ∪ 𝑗 = ∪ ran 𝑓) → ∃𝑥 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑗 = ∪ 𝑥)
73	49, 70, 72	syl2anc 595	. . . . . . . . . . 11 ⊢ (((((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) ∧ 𝑓:𝑧⟶𝑦) ∧ ∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢)) → ∃𝑥 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑗 = ∪ 𝑥)
74	73	expl 462	. . . . . . . . . 10 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) → ((𝑓:𝑧⟶𝑦 ∧ ∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢)) → ∃𝑥 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑗 = ∪ 𝑥))
75	74	exlimdv 1960	. . . . . . . . 9 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) → (∃𝑓(𝑓:𝑧⟶𝑦 ∧ ∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢)) → ∃𝑥 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑗 = ∪ 𝑥))
76	34, 75	mpd 16	. . . . . . . 8 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) → ∃𝑥 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑗 = ∪ 𝑥)
77	76	r19.29an 3175	. . . . . . 7 ⊢ ((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ ∃𝑧 ∈ (𝒫 𝑗 ∩ Fin)𝑧Ref𝑦) → ∃𝑥 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑗 = ∪ 𝑥)
78	24, 77	impbida 812	. . . . . 6 ⊢ (((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) → (∃𝑥 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑗 = ∪ 𝑥 ↔ ∃𝑧 ∈ (𝒫 𝑗 ∩ Fin)𝑧Ref𝑦))
79	78	pm5.74da 815	. . . . 5 ⊢ ((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) → ((∪ 𝑗 = ∪ 𝑦 → ∃𝑥 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑗 = ∪ 𝑥) ↔ (∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ Fin)𝑧Ref𝑦)))
80	79	ralbidva 3192	. . . 4 ⊢ (𝑗 ∈ Top → (∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑥 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑗 = ∪ 𝑥) ↔ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ Fin)𝑧Ref𝑦)))
81	80	pm5.32i 584	. . 3 ⊢ ((𝑗 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑥 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑗 = ∪ 𝑥)) ↔ (𝑗 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ Fin)𝑧Ref𝑦)))
82		eqid 2769	. . . 4 ⊢ ∪ 𝑗 = ∪ 𝑗
83	82	iscmp 23514	. . 3 ⊢ (𝑗 ∈ Comp ↔ (𝑗 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑥 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑗 = ∪ 𝑥)))
84	82	iscref 34179	. . 3 ⊢ (𝑗 ∈ CovHasRefFin ↔ (𝑗 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ Fin)𝑧Ref𝑦)))
85	81, 83, 84	3bitr4i 306	. 2 ⊢ (𝑗 ∈ Comp ↔ 𝑗 ∈ CovHasRefFin)
86	85	eqriv 2766	1 ⊢ Comp = CovHasRefFin

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  Vcvv 3461   ∩ cin 3910   ⊆ wss 3911  𝒫 cpw 4565  ∪ cuni 4874   class class class wbr 5111  ran crn 5663   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  Fincfn 8943  Topctop 23019  Compccmp 23512  Refcref 23628  CovHasRefccref 34177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-reg 9554  ax-inf2 9610  ax-ac2 10447

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3375  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  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 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7368  df-ov 7414  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-en 8944  df-dom 8945  df-fin 8947  df-r1 9736  df-rank 9737  df-card 9925  df-ac 10100  df-cmp 23513  df-ref 23631  df-cref 34178

This theorem is referenced by:  cmpfiref  34186  cmppcmp  34193