Theorem cmpcref 34032

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 769	. . . . . . . . . . . . . . 15 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → 𝑥 ∈ (𝒫 𝑦 ∩ Fin))
2		elin 3919	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝒫 𝑦 ∩ Fin) ↔ (𝑥 ∈ 𝒫 𝑦 ∧ 𝑥 ∈ Fin))
3	1, 2	sylib 218	. . . . . . . . . . . . . 14 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → (𝑥 ∈ 𝒫 𝑦 ∧ 𝑥 ∈ Fin))
4	3	simpld 494	. . . . . . . . . . . . 13 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → 𝑥 ∈ 𝒫 𝑦)
5		elpwi 4563	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝒫 𝑦 → 𝑥 ⊆ 𝑦)
6	4, 5	syl 17	. . . . . . . . . . . 12 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → 𝑥 ⊆ 𝑦)
7		elpwi 4563	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝒫 𝑗 → 𝑦 ⊆ 𝑗)
8	7	ad4antlr 734	. . . . . . . . . . . 12 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → 𝑦 ⊆ 𝑗)
9	6, 8	sstrd 3946	. . . . . . . . . . 11 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → 𝑥 ⊆ 𝑗)
10		velpw 4561	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 𝑗 ↔ 𝑥 ⊆ 𝑗)
11	9, 10	sylibr 234	. . . . . . . . . 10 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → 𝑥 ∈ 𝒫 𝑗)
12	3	simprd 495	. . . . . . . . . 10 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → 𝑥 ∈ Fin)
13	11, 12	elind 4154	. . . . . . . . 9 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → 𝑥 ∈ (𝒫 𝑗 ∩ Fin))
14		simpr 484	. . . . . . . . . . 11 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → ∪ 𝑗 = ∪ 𝑥)
15		simpllr 776	. . . . . . . . . . 11 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → ∪ 𝑗 = ∪ 𝑦)
16	14, 15	eqtr3d 2774	. . . . . . . . . 10 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → ∪ 𝑥 = ∪ 𝑦)
17		eqid 2737	. . . . . . . . . . 11 ⊢ ∪ 𝑥 = ∪ 𝑥
18		eqid 2737	. . . . . . . . . . 11 ⊢ ∪ 𝑦 = ∪ 𝑦
19	17, 18	ssref 23471	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝒫 𝑗 ∧ 𝑥 ⊆ 𝑦 ∧ ∪ 𝑥 = ∪ 𝑦) → 𝑥Ref𝑦)
20	11, 6, 16, 19	syl3anc 1374	. . . . . . . . 9 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → 𝑥Ref𝑦)
21		breq1 5103	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑧Ref𝑦 ↔ 𝑥Ref𝑦))
22	21	rspcev 3578	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝒫 𝑗 ∩ Fin) ∧ 𝑥Ref𝑦) → ∃𝑧 ∈ (𝒫 𝑗 ∩ Fin)𝑧Ref𝑦)
23	13, 20, 22	syl2anc 585	. . . . . . . 8 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → ∃𝑧 ∈ (𝒫 𝑗 ∩ Fin)𝑧Ref𝑦)
24	23	r19.29an 3142	. . . . . . 7 ⊢ ((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ ∃𝑥 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑗 = ∪ 𝑥) → ∃𝑧 ∈ (𝒫 𝑗 ∩ Fin)𝑧Ref𝑦)
25		simplr 769	. . . . . . . . . 10 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) → 𝑧 ∈ (𝒫 𝑗 ∩ Fin))
26		vex 3446	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
27		eqid 2737	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑧 = ∪ 𝑧
28	27, 18	isref 23468	. . . . . . . . . . . . 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 497	. . . . . . . . . . 11 ⊢ (𝑧Ref𝑦 → ∀𝑢 ∈ 𝑧 ∃𝑣 ∈ 𝑦 𝑢 ⊆ 𝑣)
31	30	adantl 481	. . . . . . . . . 10 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) → ∀𝑢 ∈ 𝑧 ∃𝑣 ∈ 𝑦 𝑢 ⊆ 𝑣)
32		sseq2 3962	. . . . . . . . . . 11 ⊢ (𝑣 = (𝑓‘𝑢) → (𝑢 ⊆ 𝑣 ↔ 𝑢 ⊆ (𝑓‘𝑢)))
33	32	ac6sg 10410	. . . . . . . . . 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 65	. . . . . . . . 9 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) → ∃𝑓(𝑓:𝑧⟶𝑦 ∧ ∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢)))
35		simplr 769	. . . . . . . . . . . . . . 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 6678	. . . . . . . . . . . . . 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 3446	. . . . . . . . . . . . . . . 16 ⊢ 𝑓 ∈ V
38	37	rnex 7862	. . . . . . . . . . . . . . 15 ⊢ ran 𝑓 ∈ V
39	38	elpw 4560	. . . . . . . . . . . . . 14 ⊢ (ran 𝑓 ∈ 𝒫 𝑦 ↔ ran 𝑓 ⊆ 𝑦)
40	36, 39	sylibr 234	. . . . . . . . . . . . 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 6671	. . . . . . . . . . . . . . 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 3919	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ (𝒫 𝑗 ∩ Fin) ↔ (𝑧 ∈ 𝒫 𝑗 ∧ 𝑧 ∈ Fin))
43	42	simprbi 497	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝒫 𝑗 ∩ Fin) → 𝑧 ∈ Fin)
44	43	ad4antlr 734	. . . . . . . . . . . . . . 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 9114	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 Fn 𝑧 ∧ 𝑧 ∈ Fin) → 𝑓 ∈ Fin)
46	41, 44, 45	syl2anc 585	. . . . . . . . . . . . . 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 9252	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ Fin → ran 𝑓 ∈ Fin)
48	46, 47	syl 17	. . . . . . . . . . . . 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 4154	. . . . . . . . . . . 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 786	. . . . . . . . . . . . 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 23469	. . . . . . . . . . . . . . . 16 ⊢ (𝑧Ref𝑦 → ∪ 𝑦 = ∪ 𝑧)
52	51	ad3antlr 732	. . . . . . . . . . . . . . 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 1916	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑢(((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) ∧ 𝑓:𝑧⟶𝑦)
54		nfra1 3262	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑢∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢)
55	53, 54	nfan 1901	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑢((((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) ∧ 𝑓:𝑧⟶𝑦) ∧ ∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢))
56		rspa 3227	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢) ∧ 𝑢 ∈ 𝑧) → 𝑢 ⊆ (𝑓‘𝑢))
57	56	adantll 715	. . . . . . . . . . . . . . . . . . . 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 3934	. . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . 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 3240	. . . . . . . . . . . . . . . . 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 4869	. . . . . . . . . . . . . . . . . 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 7209	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 Fn 𝑧 → (𝑥 ∈ ∪ ran 𝑓 ↔ ∃𝑢 ∈ 𝑧 𝑥 ∈ (𝑓‘𝑢)))
64	41, 63	syl 17	. . . . . . . . . . . . . . . . 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 294	. . . . . . . . . . . . . . . 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 3941	. . . . . . . . . . . . . . 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 3970	. . . . . . . . . . . . . 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 4875	. . . . . . . . . . . . . 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 3953	. . . . . . . . . . . . 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 2772	. . . . . . . . . . . 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 4876	. . . . . . . . . . . . 13 ⊢ (𝑥 = ran 𝑓 → ∪ 𝑥 = ∪ ran 𝑓)
72	71	rspceeqv 3601	. . . . . . . . . . . 12 ⊢ ((ran 𝑓 ∈ (𝒫 𝑦 ∩ Fin) ∧ ∪ 𝑗 = ∪ ran 𝑓) → ∃𝑥 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑗 = ∪ 𝑥)
73	49, 70, 72	syl2anc 585	. . . . . . . . . . 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 457	. . . . . . . . . 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 1935	. . . . . . . . 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 15	. . . . . . . 8 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) → ∃𝑥 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑗 = ∪ 𝑥)
77	76	r19.29an 3142	. . . . . . 7 ⊢ ((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ ∃𝑧 ∈ (𝒫 𝑗 ∩ Fin)𝑧Ref𝑦) → ∃𝑥 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑗 = ∪ 𝑥)
78	24, 77	impbida 801	. . . . . 6 ⊢ (((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) → (∃𝑥 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑗 = ∪ 𝑥 ↔ ∃𝑧 ∈ (𝒫 𝑗 ∩ Fin)𝑧Ref𝑦))
79	78	pm5.74da 804	. . . . 5 ⊢ ((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) → ((∪ 𝑗 = ∪ 𝑦 → ∃𝑥 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑗 = ∪ 𝑥) ↔ (∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ Fin)𝑧Ref𝑦)))
80	79	ralbidva 3159	. . . 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 574	. . 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 2737	. . . 4 ⊢ ∪ 𝑗 = ∪ 𝑗
83	82	iscmp 23347	. . 3 ⊢ (𝑗 ∈ Comp ↔ (𝑗 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑥 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑗 = ∪ 𝑥)))
84	82	iscref 34026	. . 3 ⊢ (𝑗 ∈ CovHasRefFin ↔ (𝑗 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ Fin)𝑧Ref𝑦)))
85	81, 83, 84	3bitr4i 303	. 2 ⊢ (𝑗 ∈ Comp ↔ 𝑗 ∈ CovHasRefFin)
86	85	eqriv 2734	1 ⊢ Comp = CovHasRefFin

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  Vcvv 3442   ∩ cin 3902   ⊆ wss 3903  𝒫 cpw 4556  ∪ cuni 4865   class class class wbr 5100  ran crn 5633   Fn wfn 6495  ⟶wf 6496  ‘cfv 6500  Fincfn 8895  Topctop 22852  Compccmp 23345  Refcref 23461  CovHasRefccref 34024

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-reg 9509  ax-inf2 9562  ax-ac2 10385

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-en 8896  df-dom 8897  df-fin 8899  df-r1 9688  df-rank 9689  df-card 9863  df-ac 10038  df-cmp 23346  df-ref 23464  df-cref 34025

This theorem is referenced by:  cmpfiref  34033  cmppcmp  34040