Theorem cmpcref 34043

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 774	. . . . . . . . . . . . . . 15 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → 𝑥 ∈ (𝒫 𝑦 ∩ Fin))
2		elin 3899	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝒫 𝑦 ∩ Fin) ↔ (𝑥 ∈ 𝒫 𝑦 ∧ 𝑥 ∈ Fin))
3	1, 2	sylib 219	. . . . . . . . . . . . . 14 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → (𝑥 ∈ 𝒫 𝑦 ∧ 𝑥 ∈ Fin))
4	3	simpld 495	. . . . . . . . . . . . 13 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → 𝑥 ∈ 𝒫 𝑦)
5		elpwi 4537	. . . . . . . . . . . . 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 4537	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝒫 𝑗 → 𝑦 ⊆ 𝑗)
8	7	ad4antlr 739	. . . . . . . . . . . 12 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → 𝑦 ⊆ 𝑗)
9	6, 8	sstrd 3925	. . . . . . . . . . 11 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → 𝑥 ⊆ 𝑗)
10		velpw 4535	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 𝑗 ↔ 𝑥 ⊆ 𝑗)
11	9, 10	sylibr 235	. . . . . . . . . 10 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → 𝑥 ∈ 𝒫 𝑗)
12	3	simprd 496	. . . . . . . . . 10 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → 𝑥 ∈ Fin)
13	11, 12	elind 4130	. . . . . . . . 9 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → 𝑥 ∈ (𝒫 𝑗 ∩ Fin))
14		simpr 485	. . . . . . . . . . 11 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → ∪ 𝑗 = ∪ 𝑥)
15		simpllr 781	. . . . . . . . . . 11 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → ∪ 𝑗 = ∪ 𝑦)
16	14, 15	eqtr3d 2776	. . . . . . . . . 10 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → ∪ 𝑥 = ∪ 𝑦)
17		eqid 2739	. . . . . . . . . . 11 ⊢ ∪ 𝑥 = ∪ 𝑥
18		eqid 2739	. . . . . . . . . . 11 ⊢ ∪ 𝑦 = ∪ 𝑦
19	17, 18	ssref 23496	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝒫 𝑗 ∧ 𝑥 ⊆ 𝑦 ∧ ∪ 𝑥 = ∪ 𝑦) → 𝑥Ref𝑦)
20	11, 6, 16, 19	syl3anc 1379	. . . . . . . . 9 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → 𝑥Ref𝑦)
21		breq1 5076	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑧Ref𝑦 ↔ 𝑥Ref𝑦))
22	21	rspcev 3560	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝒫 𝑗 ∩ Fin) ∧ 𝑥Ref𝑦) → ∃𝑧 ∈ (𝒫 𝑗 ∩ Fin)𝑧Ref𝑦)
23	13, 20, 22	syl2anc 590	. . . . . . . 8 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → ∃𝑧 ∈ (𝒫 𝑗 ∩ Fin)𝑧Ref𝑦)
24	23	r19.29an 3143	. . . . . . 7 ⊢ ((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ ∃𝑥 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑗 = ∪ 𝑥) → ∃𝑧 ∈ (𝒫 𝑗 ∩ Fin)𝑧Ref𝑦)
25		simplr 774	. . . . . . . . . 10 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) → 𝑧 ∈ (𝒫 𝑗 ∩ Fin))
26		vex 3435	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
27		eqid 2739	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑧 = ∪ 𝑧
28	27, 18	isref 23493	. . . . . . . . . . . . 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 498	. . . . . . . . . . 11 ⊢ (𝑧Ref𝑦 → ∀𝑢 ∈ 𝑧 ∃𝑣 ∈ 𝑦 𝑢 ⊆ 𝑣)
31	30	adantl 482	. . . . . . . . . 10 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) → ∀𝑢 ∈ 𝑧 ∃𝑣 ∈ 𝑦 𝑢 ⊆ 𝑣)
32		sseq2 3941	. . . . . . . . . . 11 ⊢ (𝑣 = (𝑓‘𝑢) → (𝑢 ⊆ 𝑣 ↔ 𝑢 ⊆ (𝑓‘𝑢)))
33	32	ac6sg 10402	. . . . . . . . . 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 774	. . . . . . . . . . . . . . 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 6664	. . . . . . . . . . . . . 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 3435	. . . . . . . . . . . . . . . 16 ⊢ 𝑓 ∈ V
38	37	rnex 7851	. . . . . . . . . . . . . . 15 ⊢ ran 𝑓 ∈ V
39	38	elpw 4534	. . . . . . . . . . . . . 14 ⊢ (ran 𝑓 ∈ 𝒫 𝑦 ↔ ran 𝑓 ⊆ 𝑦)
40	36, 39	sylibr 235	. . . . . . . . . . . . 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 6657	. . . . . . . . . . . . . . 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 3899	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ (𝒫 𝑗 ∩ Fin) ↔ (𝑧 ∈ 𝒫 𝑗 ∧ 𝑧 ∈ Fin))
43	42	simprbi 498	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝒫 𝑗 ∩ Fin) → 𝑧 ∈ Fin)
44	43	ad4antlr 739	. . . . . . . . . . . . . . 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 9103	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 Fn 𝑧 ∧ 𝑧 ∈ Fin) → 𝑓 ∈ Fin)
46	41, 44, 45	syl2anc 590	. . . . . . . . . . . . . 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 9241	. . . . . . . . . . . . . 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 4130	. . . . . . . . . . . 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 791	. . . . . . . . . . . . 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 23494	. . . . . . . . . . . . . . . 16 ⊢ (𝑧Ref𝑦 → ∪ 𝑦 = ∪ 𝑧)
52	51	ad3antlr 737	. . . . . . . . . . . . . . 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 1921	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑢(((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) ∧ 𝑓:𝑧⟶𝑦)
54		nfra1 3263	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑢∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢)
55	53, 54	nfan 1906	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑢((((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) ∧ 𝑓:𝑧⟶𝑦) ∧ ∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢))
56		rspa 3228	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢) ∧ 𝑢 ∈ 𝑧) → 𝑢 ⊆ (𝑓‘𝑢))
57	56	adantll 720	. . . . . . . . . . . . . . . . . . . 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 3914	. . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . 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 3241	. . . . . . . . . . . . . . . . 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 4843	. . . . . . . . . . . . . . . . . 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 7198	. . . . . . . . . . . . . . . . . 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 295	. . . . . . . . . . . . . . . 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 3921	. . . . . . . . . . . . . . 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 3949	. . . . . . . . . . . . . 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 4849	. . . . . . . . . . . . . 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 3932	. . . . . . . . . . . . 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 2774	. . . . . . . . . . . 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 4850	. . . . . . . . . . . . 13 ⊢ (𝑥 = ran 𝑓 → ∪ 𝑥 = ∪ ran 𝑓)
72	71	rspceeqv 3583	. . . . . . . . . . . 12 ⊢ ((ran 𝑓 ∈ (𝒫 𝑦 ∩ Fin) ∧ ∪ 𝑗 = ∪ ran 𝑓) → ∃𝑥 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑗 = ∪ 𝑥)
73	49, 70, 72	syl2anc 590	. . . . . . . . . . 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 458	. . . . . . . . . 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 1940	. . . . . . . . 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 3143	. . . . . . 7 ⊢ ((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ ∃𝑧 ∈ (𝒫 𝑗 ∩ Fin)𝑧Ref𝑦) → ∃𝑥 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑗 = ∪ 𝑥)
78	24, 77	impbida 806	. . . . . 6 ⊢ (((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) → (∃𝑥 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑗 = ∪ 𝑥 ↔ ∃𝑧 ∈ (𝒫 𝑗 ∩ Fin)𝑧Ref𝑦))
79	78	pm5.74da 809	. . . . 5 ⊢ ((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) → ((∪ 𝑗 = ∪ 𝑦 → ∃𝑥 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑗 = ∪ 𝑥) ↔ (∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ Fin)𝑧Ref𝑦)))
80	79	ralbidva 3160	. . . 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 579	. . 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 2739	. . . 4 ⊢ ∪ 𝑗 = ∪ 𝑗
83	82	iscmp 23372	. . 3 ⊢ (𝑗 ∈ Comp ↔ (𝑗 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑥 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑗 = ∪ 𝑥)))
84	82	iscref 34037	. . 3 ⊢ (𝑗 ∈ CovHasRefFin ↔ (𝑗 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ Fin)𝑧Ref𝑦)))
85	81, 83, 84	3bitr4i 304	. 2 ⊢ (𝑗 ∈ Comp ↔ 𝑗 ∈ CovHasRefFin)
86	85	eqriv 2736	1 ⊢ Comp = CovHasRefFin

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547  ∃wex 1786   ∈ wcel 2119  ∀wral 3053  ∃wrex 3063  Vcvv 3431   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4530  ∪ cuni 4839   class class class wbr 5073  ran crn 5620   Fn wfn 6481  ⟶wf 6482  ‘cfv 6486  Fincfn 8884  Topctop 22877  Compccmp 23370  Refcref 23486  CovHasRefccref 34035

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679  ax-reg 9498  ax-inf2 9554  ax-ac2 10377

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-int 4879  df-iun 4924  df-iin 4925  df-br 5074  df-opab 5136  df-mpt 5155  df-tr 5181  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7314  df-ov 7360  df-om 7808  df-1st 7932  df-2nd 7933  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-en 8885  df-dom 8886  df-fin 8888  df-r1 9680  df-rank 9681  df-card 9855  df-ac 10030  df-cmp 23371  df-ref 23489  df-cref 34036

This theorem is referenced by:  cmpfiref  34044  cmppcmp  34051