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Theorem cncmp 22896

Description: Compactness is respected by a continuous onto map. (Contributed by Jeff Hankins, 12-Jul-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)

**Hypothesis**
Ref	Expression
cncmp.2	⊢ 𝑌 = ∪ 𝐾

**Assertion**
Ref	Expression
cncmp	⊢ ((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Comp)

Proof of Theorem cncmp Dummy variables 𝑐 𝑑 𝑠 𝑢 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cntop2 22745	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
2	1	3ad2ant3 1136	. 2 ⊢ ((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
3		elpwi 4610	. . . 4 ⊢ (𝑢 ∈ 𝒫 𝐾 → 𝑢 ⊆ 𝐾)
4		simpl1 1192	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → 𝐽 ∈ Comp)
5		simpl3 1194	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → 𝐹 ∈ (𝐽 Cn 𝐾))
6		simprl 770	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → 𝑢 ⊆ 𝐾)
7	6	sselda 3983	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑦 ∈ 𝑢) → 𝑦 ∈ 𝐾)
8		cnima 22769	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 ∈ 𝐾) → (^◡𝐹 “ 𝑦) ∈ 𝐽)
9	5, 7, 8	syl2an2r 684	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑦 ∈ 𝑢) → (^◡𝐹 “ 𝑦) ∈ 𝐽)
10	9	fmpttd 7115	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)):𝑢⟶𝐽)
11	10	frnd 6726	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ⊆ 𝐽)
12		simprr 772	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → 𝑌 = ∪ 𝑢)
13	12	imaeq2d 6060	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → (^◡𝐹 “ 𝑌) = (^◡𝐹 “ ∪ 𝑢))
14		eqid 2733	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
15		cncmp.2	. . . . . . . . . . 11 ⊢ 𝑌 = ∪ 𝐾
16	14, 15	cnf 22750	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶𝑌)
17	5, 16	syl 17	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → 𝐹:∪ 𝐽⟶𝑌)
18		fimacnv 6740	. . . . . . . . 9 ⊢ (𝐹:∪ 𝐽⟶𝑌 → (^◡𝐹 “ 𝑌) = ∪ 𝐽)
19	17, 18	syl 17	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → (^◡𝐹 “ 𝑌) = ∪ 𝐽)
20	9	ralrimiva 3147	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → ∀𝑦 ∈ 𝑢 (^◡𝐹 “ 𝑦) ∈ 𝐽)
21		dfiun2g 5034	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑢 (^◡𝐹 “ 𝑦) ∈ 𝐽 → ∪ 𝑦 ∈ 𝑢 (^◡𝐹 “ 𝑦) = ∪ {𝑥 ∣ ∃𝑦 ∈ 𝑢 𝑥 = (^◡𝐹 “ 𝑦)})
22	20, 21	syl 17	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → ∪ 𝑦 ∈ 𝑢 (^◡𝐹 “ 𝑦) = ∪ {𝑥 ∣ ∃𝑦 ∈ 𝑢 𝑥 = (^◡𝐹 “ 𝑦)})
23		imauni 7245	. . . . . . . . 9 ⊢ (^◡𝐹 “ ∪ 𝑢) = ∪ 𝑦 ∈ 𝑢 (^◡𝐹 “ 𝑦)
24		eqid 2733	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) = (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦))
25	24	rnmpt 5955	. . . . . . . . . 10 ⊢ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) = {𝑥 ∣ ∃𝑦 ∈ 𝑢 𝑥 = (^◡𝐹 “ 𝑦)}
26	25	unieqi 4922	. . . . . . . . 9 ⊢ ∪ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) = ∪ {𝑥 ∣ ∃𝑦 ∈ 𝑢 𝑥 = (^◡𝐹 “ 𝑦)}
27	22, 23, 26	3eqtr4g 2798	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → (^◡𝐹 “ ∪ 𝑢) = ∪ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)))
28	13, 19, 27	3eqtr3d 2781	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → ∪ 𝐽 = ∪ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)))
29	14	cmpcov 22893	. . . . . . 7 ⊢ ((𝐽 ∈ Comp ∧ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦))) → ∃𝑠 ∈ (𝒫 ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ∩ Fin)∪ 𝐽 = ∪ 𝑠)
30	4, 11, 28, 29	syl3anc 1372	. . . . . 6 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → ∃𝑠 ∈ (𝒫 ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ∩ Fin)∪ 𝐽 = ∪ 𝑠)
31		elfpw 9354	. . . . . . . 8 ⊢ (𝑠 ∈ (𝒫 ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ∩ Fin) ↔ (𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin))
32		simprll 778	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑠)) → 𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)))
33	32	sselda 3983	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑠)) ∧ 𝑐 ∈ 𝑠) → 𝑐 ∈ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)))
34		simpll2 1214	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑦 ∈ 𝑢) → 𝐹:𝑋–onto→𝑌)
35		elssuni 4942	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ 𝐾 → 𝑦 ⊆ ∪ 𝐾)
36	35, 15	sseqtrrdi 4034	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ 𝐾 → 𝑦 ⊆ 𝑌)
37	7, 36	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑦 ∈ 𝑢) → 𝑦 ⊆ 𝑌)
38		foimacnv 6851	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑦 ⊆ 𝑌) → (𝐹 “ (^◡𝐹 “ 𝑦)) = 𝑦)
39	34, 37, 38	syl2anc 585	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑦 ∈ 𝑢) → (𝐹 “ (^◡𝐹 “ 𝑦)) = 𝑦)
40		simpr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑦 ∈ 𝑢) → 𝑦 ∈ 𝑢)
41	39, 40	eqeltrd 2834	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑦 ∈ 𝑢) → (𝐹 “ (^◡𝐹 “ 𝑦)) ∈ 𝑢)
42	41	ralrimiva 3147	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → ∀𝑦 ∈ 𝑢 (𝐹 “ (^◡𝐹 “ 𝑦)) ∈ 𝑢)
43		imaeq2 6056	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 = (^◡𝐹 “ 𝑦) → (𝐹 “ 𝑐) = (𝐹 “ (^◡𝐹 “ 𝑦)))
44	43	eleq1d 2819	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 = (^◡𝐹 “ 𝑦) → ((𝐹 “ 𝑐) ∈ 𝑢 ↔ (𝐹 “ (^◡𝐹 “ 𝑦)) ∈ 𝑢))
45	24, 44	ralrnmptw 7096	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑦 ∈ 𝑢 (^◡𝐹 “ 𝑦) ∈ 𝐽 → (∀𝑐 ∈ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦))(𝐹 “ 𝑐) ∈ 𝑢 ↔ ∀𝑦 ∈ 𝑢 (𝐹 “ (^◡𝐹 “ 𝑦)) ∈ 𝑢))
46	20, 45	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → (∀𝑐 ∈ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦))(𝐹 “ 𝑐) ∈ 𝑢 ↔ ∀𝑦 ∈ 𝑢 (𝐹 “ (^◡𝐹 “ 𝑦)) ∈ 𝑢))
47	42, 46	mpbird 257	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → ∀𝑐 ∈ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦))(𝐹 “ 𝑐) ∈ 𝑢)
48	47	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑠)) → ∀𝑐 ∈ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦))(𝐹 “ 𝑐) ∈ 𝑢)
49	48	r19.21bi 3249	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑠)) ∧ 𝑐 ∈ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦))) → (𝐹 “ 𝑐) ∈ 𝑢)
50	33, 49	syldan 592	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑠)) ∧ 𝑐 ∈ 𝑠) → (𝐹 “ 𝑐) ∈ 𝑢)
51	50	fmpttd 7115	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑠)) → (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)):𝑠⟶𝑢)
52	51	frnd 6726	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑠)) → ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) ⊆ 𝑢)
53		simprlr 779	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑠)) → 𝑠 ∈ Fin)
54		eqid 2733	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) = (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐))
55	54	rnmpt 5955	. . . . . . . . . . . . 13 ⊢ ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) = {𝑑 ∣ ∃𝑐 ∈ 𝑠 𝑑 = (𝐹 “ 𝑐)}
56		abrexfi 9352	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ Fin → {𝑑 ∣ ∃𝑐 ∈ 𝑠 𝑑 = (𝐹 “ 𝑐)} ∈ Fin)
57	55, 56	eqeltrid 2838	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ Fin → ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) ∈ Fin)
58	53, 57	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑠)) → ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) ∈ Fin)
59		elfpw 9354	. . . . . . . . . . 11 ⊢ (ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) ∈ (𝒫 𝑢 ∩ Fin) ↔ (ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) ⊆ 𝑢 ∧ ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) ∈ Fin))
60	52, 58, 59	sylanbrc 584	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑠)) → ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) ∈ (𝒫 𝑢 ∩ Fin))
61	17	adantr 482	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑠)) → 𝐹:∪ 𝐽⟶𝑌)
62	61	fdmd 6729	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑠)) → dom 𝐹 = ∪ 𝐽)
63		simpll2 1214	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑠)) → 𝐹:𝑋–onto→𝑌)
64		fof 6806	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌)
65		fdm 6727	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋)
66	63, 64, 65	3syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑠)) → dom 𝐹 = 𝑋)
67		simprr 772	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑠)) → ∪ 𝐽 = ∪ 𝑠)
68	62, 66, 67	3eqtr3d 2781	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑠)) → 𝑋 = ∪ 𝑠)
69	68	imaeq2d 6060	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑠)) → (𝐹 “ 𝑋) = (𝐹 “ ∪ 𝑠))
70		foima 6811	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋–onto→𝑌 → (𝐹 “ 𝑋) = 𝑌)
71	63, 70	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑠)) → (𝐹 “ 𝑋) = 𝑌)
72	50	ralrimiva 3147	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑠)) → ∀𝑐 ∈ 𝑠 (𝐹 “ 𝑐) ∈ 𝑢)
73		dfiun2g 5034	. . . . . . . . . . . . 13 ⊢ (∀𝑐 ∈ 𝑠 (𝐹 “ 𝑐) ∈ 𝑢 → ∪ 𝑐 ∈ 𝑠 (𝐹 “ 𝑐) = ∪ {𝑑 ∣ ∃𝑐 ∈ 𝑠 𝑑 = (𝐹 “ 𝑐)})
74	72, 73	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑠)) → ∪ 𝑐 ∈ 𝑠 (𝐹 “ 𝑐) = ∪ {𝑑 ∣ ∃𝑐 ∈ 𝑠 𝑑 = (𝐹 “ 𝑐)})
75		imauni 7245	. . . . . . . . . . . 12 ⊢ (𝐹 “ ∪ 𝑠) = ∪ 𝑐 ∈ 𝑠 (𝐹 “ 𝑐)
76	55	unieqi 4922	. . . . . . . . . . . 12 ⊢ ∪ ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) = ∪ {𝑑 ∣ ∃𝑐 ∈ 𝑠 𝑑 = (𝐹 “ 𝑐)}
77	74, 75, 76	3eqtr4g 2798	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑠)) → (𝐹 “ ∪ 𝑠) = ∪ ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)))
78	69, 71, 77	3eqtr3d 2781	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑠)) → 𝑌 = ∪ ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)))
79		unieq 4920	. . . . . . . . . . 11 ⊢ (𝑣 = ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) → ∪ 𝑣 = ∪ ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)))
80	79	rspceeqv 3634	. . . . . . . . . 10 ⊢ ((ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) ∈ (𝒫 𝑢 ∩ Fin) ∧ 𝑌 = ∪ ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐))) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = ∪ 𝑣)
81	60, 78, 80	syl2anc 585	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑠)) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = ∪ 𝑣)
82	81	expr 458	. . . . . . . 8 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ (𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin)) → (∪ 𝐽 = ∪ 𝑠 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = ∪ 𝑣))
83	31, 82	sylan2b 595	. . . . . . 7 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑠 ∈ (𝒫 ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ∩ Fin)) → (∪ 𝐽 = ∪ 𝑠 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = ∪ 𝑣))
84	83	rexlimdva 3156	. . . . . 6 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → (∃𝑠 ∈ (𝒫 ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ∩ Fin)∪ 𝐽 = ∪ 𝑠 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = ∪ 𝑣))
85	30, 84	mpd 15	. . . . 5 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = ∪ 𝑣)
86	85	expr 458	. . . 4 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑢 ⊆ 𝐾) → (𝑌 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = ∪ 𝑣))
87	3, 86	sylan2 594	. . 3 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑢 ∈ 𝒫 𝐾) → (𝑌 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = ∪ 𝑣))
88	87	ralrimiva 3147	. 2 ⊢ ((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑢 ∈ 𝒫 𝐾(𝑌 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = ∪ 𝑣))
89	15	iscmp 22892	. 2 ⊢ (𝐾 ∈ Comp ↔ (𝐾 ∈ Top ∧ ∀𝑢 ∈ 𝒫 𝐾(𝑌 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = ∪ 𝑣)))
90	2, 88, 89	sylanbrc 584	1 ⊢ ((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Comp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 {cab 2710 ∀wral 3062 ∃wrex 3071 ∩ cin 3948 ⊆ wss 3949 𝒫 cpw 4603 ∪ cuni 4909 ∪ ciun 4998 ↦ cmpt 5232 ^◡ccnv 5676 dom cdm 5677 ran crn 5678 “ cima 5680 ⟶wf 6540 –onto→wfo 6542 (class class class)co 7409 Fincfn 8939 Topctop 22395 Cn ccn 22728 Compccmp 22890

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-1o 8466 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-fin 8943 df-top 22396 df-topon 22413 df-cn 22731 df-cmp 22891

This theorem is referenced by: rncmp 22900 txcmpb 23148 qtopcmp 23212 cmphmph 23292

W3C validator