Theorem cmpcref 32431

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 767	. . . . . . . . . . . . . . 15 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → 𝑥 ∈ (𝒫 𝑦 ∩ Fin))
2		elin 3926	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝒫 𝑦 ∩ Fin) ↔ (𝑥 ∈ 𝒫 𝑦 ∧ 𝑥 ∈ Fin))
3	1, 2	sylib 217	. . . . . . . . . . . . . 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 4567	. . . . . . . . . . . . 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 4567	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝒫 𝑗 → 𝑦 ⊆ 𝑗)
8	7	ad4antlr 731	. . . . . . . . . . . 12 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → 𝑦 ⊆ 𝑗)
9	6, 8	sstrd 3954	. . . . . . . . . . 11 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → 𝑥 ⊆ 𝑗)
10		velpw 4565	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 𝑗 ↔ 𝑥 ⊆ 𝑗)
11	9, 10	sylibr 233	. . . . . . . . . 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 4154	. . . . . . . . 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 774	. . . . . . . . . . 11 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → ∪ 𝑗 = ∪ 𝑦)
16	14, 15	eqtr3d 2778	. . . . . . . . . 10 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → ∪ 𝑥 = ∪ 𝑦)
17		eqid 2736	. . . . . . . . . . 11 ⊢ ∪ 𝑥 = ∪ 𝑥
18		eqid 2736	. . . . . . . . . . 11 ⊢ ∪ 𝑦 = ∪ 𝑦
19	17, 18	ssref 22863	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝒫 𝑗 ∧ 𝑥 ⊆ 𝑦 ∧ ∪ 𝑥 = ∪ 𝑦) → 𝑥Ref𝑦)
20	11, 6, 16, 19	syl3anc 1371	. . . . . . . . 9 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → 𝑥Ref𝑦)
21		breq1 5108	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑧Ref𝑦 ↔ 𝑥Ref𝑦))
22	21	rspcev 3581	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝒫 𝑗 ∩ Fin) ∧ 𝑥Ref𝑦) → ∃𝑧 ∈ (𝒫 𝑗 ∩ Fin)𝑧Ref𝑦)
23	13, 20, 22	syl2anc 584	. . . . . . . 8 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑥 ∈ (𝒫 𝑦 ∩ Fin)) ∧ ∪ 𝑗 = ∪ 𝑥) → ∃𝑧 ∈ (𝒫 𝑗 ∩ Fin)𝑧Ref𝑦)
24	23	r19.29an 3155	. . . . . . 7 ⊢ ((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ ∃𝑥 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑗 = ∪ 𝑥) → ∃𝑧 ∈ (𝒫 𝑗 ∩ Fin)𝑧Ref𝑦)
25		simplr 767	. . . . . . . . . 10 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) → 𝑧 ∈ (𝒫 𝑗 ∩ Fin))
26		vex 3449	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
27		eqid 2736	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑧 = ∪ 𝑧
28	27, 18	isref 22860	. . . . . . . . . . . . 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 482	. . . . . . . . . 10 ⊢ (((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) → ∀𝑢 ∈ 𝑧 ∃𝑣 ∈ 𝑦 𝑢 ⊆ 𝑣)
32		sseq2 3970	. . . . . . . . . . 11 ⊢ (𝑣 = (𝑓‘𝑢) → (𝑢 ⊆ 𝑣 ↔ 𝑢 ⊆ (𝑓‘𝑢)))
33	32	ac6sg 10424	. . . . . . . . . 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 767	. . . . . . . . . . . . . . 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 6676	. . . . . . . . . . . . . 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 3449	. . . . . . . . . . . . . . . 16 ⊢ 𝑓 ∈ V
38	37	rnex 7849	. . . . . . . . . . . . . . 15 ⊢ ran 𝑓 ∈ V
39	38	elpw 4564	. . . . . . . . . . . . . 14 ⊢ (ran 𝑓 ∈ 𝒫 𝑦 ↔ ran 𝑓 ⊆ 𝑦)
40	36, 39	sylibr 233	. . . . . . . . . . . . 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 6669	. . . . . . . . . . . . . . 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 3926	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ (𝒫 𝑗 ∩ Fin) ↔ (𝑧 ∈ 𝒫 𝑗 ∧ 𝑧 ∈ Fin))
43	42	simprbi 497	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝒫 𝑗 ∩ Fin) → 𝑧 ∈ Fin)
44	43	ad4antlr 731	. . . . . . . . . . . . . . 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 9125	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 Fn 𝑧 ∧ 𝑧 ∈ Fin) → 𝑓 ∈ Fin)
46	41, 44, 45	syl2anc 584	. . . . . . . . . . . . . 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 9279	. . . . . . . . . . . . . 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 784	. . . . . . . . . . . . 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 22861	. . . . . . . . . . . . . . . 16 ⊢ (𝑧Ref𝑦 → ∪ 𝑦 = ∪ 𝑧)
52	51	ad3antlr 729	. . . . . . . . . . . . . . 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 1917	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑢(((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) ∧ 𝑓:𝑧⟶𝑦)
54		nfra1 3267	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑢∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢)
55	53, 54	nfan 1902	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑢((((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ 𝑧 ∈ (𝒫 𝑗 ∩ Fin)) ∧ 𝑧Ref𝑦) ∧ 𝑓:𝑧⟶𝑦) ∧ ∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢))
56		rspa 3231	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((∀𝑢 ∈ 𝑧 𝑢 ⊆ (𝑓‘𝑢) ∧ 𝑢 ∈ 𝑧) → 𝑢 ⊆ (𝑓‘𝑢))
57	56	adantll 712	. . . . . . . . . . . . . . . . . . . 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 3943	. . . . . . . . . . . . . . . . . . 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 3244	. . . . . . . . . . . . . . . . 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 7201	. . . . . . . . . . . . . . . . . 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 293	. . . . . . . . . . . . . . . 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 3950	. . . . . . . . . . . . . . 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 3982	. . . . . . . . . . . . . 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 3961	. . . . . . . . . . . . 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 2776	. . . . . . . . . . . 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 3595	. . . . . . . . . . . 12 ⊢ ((ran 𝑓 ∈ (𝒫 𝑦 ∩ Fin) ∧ ∪ 𝑗 = ∪ ran 𝑓) → ∃𝑥 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑗 = ∪ 𝑥)
73	49, 70, 72	syl2anc 584	. . . . . . . . . . 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 1936	. . . . . . . . 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 3155	. . . . . . 7 ⊢ ((((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) ∧ ∃𝑧 ∈ (𝒫 𝑗 ∩ Fin)𝑧Ref𝑦) → ∃𝑥 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑗 = ∪ 𝑥)
78	24, 77	impbida 799	. . . . . 6 ⊢ (((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) ∧ ∪ 𝑗 = ∪ 𝑦) → (∃𝑥 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑗 = ∪ 𝑥 ↔ ∃𝑧 ∈ (𝒫 𝑗 ∩ Fin)𝑧Ref𝑦))
79	78	pm5.74da 802	. . . . 5 ⊢ ((𝑗 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑗) → ((∪ 𝑗 = ∪ 𝑦 → ∃𝑥 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑗 = ∪ 𝑥) ↔ (∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ Fin)𝑧Ref𝑦)))
80	79	ralbidva 3172	. . . 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 575	. . 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 2736	. . . 4 ⊢ ∪ 𝑗 = ∪ 𝑗
83	82	iscmp 22739	. . 3 ⊢ (𝑗 ∈ Comp ↔ (𝑗 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑥 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑗 = ∪ 𝑥)))
84	82	iscref 32425	. . 3 ⊢ (𝑗 ∈ CovHasRefFin ↔ (𝑗 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ Fin)𝑧Ref𝑦)))
85	81, 83, 84	3bitr4i 302	. 2 ⊢ (𝑗 ∈ Comp ↔ 𝑗 ∈ CovHasRefFin)
86	85	eqriv 2733	1 ⊢ Comp = CovHasRefFin

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∀wral 3064  ∃wrex 3073  Vcvv 3445   ∩ cin 3909   ⊆ wss 3910  𝒫 cpw 4560  ∪ cuni 4865   class class class wbr 5105  ran crn 5634   Fn wfn 6491  ⟶wf 6492  ‘cfv 6496  Fincfn 8883  Topctop 22242  Compccmp 22737  Refcref 22853  CovHasRefccref 32423

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-reg 9528  ax-inf2 9577  ax-ac2 10399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-fin 8887  df-r1 9700  df-rank 9701  df-card 9875  df-ac 10052  df-cmp 22738  df-ref 22856  df-cref 32424

This theorem is referenced by:  cmpfiref  32432  cmppcmp  32439