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Theorem cncmp 23421

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 23270	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
2	1	3ad2ant3 1144	. 2 ⊢ ((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
3		elpwi 4552	. . . 4 ⊢ (𝑢 ∈ 𝒫 𝐾 → 𝑢 ⊆ 𝐾)
4		simpl1 1201	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → 𝐽 ∈ Comp)
5		simpl3 1203	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → 𝐹 ∈ (𝐽 Cn 𝐾))
6		simprl 778	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → 𝑢 ⊆ 𝐾)
7	6	sselda 3927	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑦 ∈ 𝑢) → 𝑦 ∈ 𝐾)
8		cnima 23294	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 ∈ 𝐾) → (^◡𝐹 “ 𝑦) ∈ 𝐽)
9	5, 7, 8	syl2an2r 693	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑦 ∈ 𝑢) → (^◡𝐹 “ 𝑦) ∈ 𝐽)
10	9	fmpttd 7081	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)):𝑢⟶𝐽)
11	10	frnd 6685	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ⊆ 𝐽)
12		simprr 780	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → 𝑌 = ∪ 𝑢)
13	12	imaeq2d 6035	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → (^◡𝐹 “ 𝑌) = (^◡𝐹 “ ∪ 𝑢))
14		eqid 2752	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
15		cncmp.2	. . . . . . . . . . 11 ⊢ 𝑌 = ∪ 𝐾
16	14, 15	cnf 23275	. . . . . . . . . 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 6699	. . . . . . . . 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 3144	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → ∀𝑦 ∈ 𝑢 (^◡𝐹 “ 𝑦) ∈ 𝐽)
21		dfiun2g 4977	. . . . . . . . . 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 7215	. . . . . . . . 9 ⊢ (^◡𝐹 “ ∪ 𝑢) = ∪ 𝑦 ∈ 𝑢 (^◡𝐹 “ 𝑦)
24		eqid 2752	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) = (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦))
25	24	rnmpt 5922	. . . . . . . . . 10 ⊢ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) = {𝑥 ∣ ∃𝑦 ∈ 𝑢 𝑥 = (^◡𝐹 “ 𝑦)}
26	25	unieqi 4867	. . . . . . . . 9 ⊢ ∪ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) = ∪ {𝑥 ∣ ∃𝑦 ∈ 𝑢 𝑥 = (^◡𝐹 “ 𝑦)}
27	22, 23, 26	3eqtr4g 2812	. . . . . . . 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 2795	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → ∪ 𝐽 = ∪ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)))
29	14	cmpcov 23418	. . . . . . 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 1382	. . . . . 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 9283	. . . . . . . 8 ⊢ (𝑠 ∈ (𝒫 ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ∩ Fin) ↔ (𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin))
32		simprll 786	. . . . . . . . . . . . . . 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 3927	. . . . . . . . . . . . . 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 1223	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑦 ∈ 𝑢) → 𝐹:𝑋–onto→𝑌)
35		elssuni 4887	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ 𝐾 → 𝑦 ⊆ ∪ 𝐾)
36	35, 15	sseqtrrdi 3968	. . . . . . . . . . . . . . . . . . . . 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 6809	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑦 ⊆ 𝑌) → (𝐹 “ (^◡𝐹 “ 𝑦)) = 𝑦)
39	34, 37, 38	syl2anc 592	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑦 ∈ 𝑢) → (𝐹 “ (^◡𝐹 “ 𝑦)) = 𝑦)
40		simpr 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑦 ∈ 𝑢) → 𝑦 ∈ 𝑢)
41	39, 40	eqeltrd 2852	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑦 ∈ 𝑢) → (𝐹 “ (^◡𝐹 “ 𝑦)) ∈ 𝑢)
42	41	ralrimiva 3144	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → ∀𝑦 ∈ 𝑢 (𝐹 “ (^◡𝐹 “ 𝑦)) ∈ 𝑢)
43		imaeq2 6031	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 = (^◡𝐹 “ 𝑦) → (𝐹 “ 𝑐) = (𝐹 “ (^◡𝐹 “ 𝑦)))
44	43	eleq1d 2837	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 = (^◡𝐹 “ 𝑦) → ((𝐹 “ 𝑐) ∈ 𝑢 ↔ (𝐹 “ (^◡𝐹 “ 𝑦)) ∈ 𝑢))
45	24, 44	ralrnmptw 7060	. . . . . . . . . . . . . . . . . 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 259	. . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . 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 3244	. . . . . . . . . . . . . 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 599	. . . . . . . . . . . . 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 7081	. . . . . . . . . . . 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 6685	. . . . . . . . . . 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 787	. . . . . . . . . . . 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 2752	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) = (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐))
55	54	rnmpt 5922	. . . . . . . . . . . . 13 ⊢ ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) = {𝑑 ∣ ∃𝑐 ∈ 𝑠 𝑑 = (𝐹 “ 𝑐)}
56		abrexfi 9281	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ Fin → {𝑑 ∣ ∃𝑐 ∈ 𝑠 𝑑 = (𝐹 “ 𝑐)} ∈ Fin)
57	55, 56	eqeltrid 2856	. . . . . . . . . . . 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 9283	. . . . . . . . . . 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 591	. . . . . . . . . 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 483	. . . . . . . . . . . . . 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 6687	. . . . . . . . . . . . 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 1223	. . . . . . . . . . . . . 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 6763	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌)
65		fdm 6686	. . . . . . . . . . . . . 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 780	. . . . . . . . . . . . 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 2795	. . . . . . . . . . . 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 6035	. . . . . . . . . . 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 6768	. . . . . . . . . . . 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 3144	. . . . . . . . . . . . 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 4977	. . . . . . . . . . . . 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 7215	. . . . . . . . . . . 12 ⊢ (𝐹 “ ∪ 𝑠) = ∪ 𝑐 ∈ 𝑠 (𝐹 “ 𝑐)
76	55	unieqi 4867	. . . . . . . . . . . 12 ⊢ ∪ ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) = ∪ {𝑑 ∣ ∃𝑐 ∈ 𝑠 𝑑 = (𝐹 “ 𝑐)}
77	74, 75, 76	3eqtr4g 2812	. . . . . . . . . . 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 2795	. . . . . . . . . 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 4866	. . . . . . . . . . 11 ⊢ (𝑣 = ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) → ∪ 𝑣 = ∪ ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)))
80	79	rspceeqv 3595	. . . . . . . . . 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 592	. . . . . . . . 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 459	. . . . . . . 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 602	. . . . . . 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 3153	. . . . . 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 459	. . . 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 601	. . 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 3144	. 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 23417	. 2 ⊢ (𝐾 ∈ Comp ↔ (𝐾 ∈ Top ∧ ∀𝑢 ∈ 𝒫 𝐾(𝑌 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = ∪ 𝑣)))
90	2, 88, 89	sylanbrc 591	1 ⊢ ((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Comp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1095 = wceq 1550 ∈ wcel 2132 {cab 2730 ∀wral 3066 ∃wrex 3076 ∩ cin 3894 ⊆ wss 3895 𝒫 cpw 4545 ∪ cuni 4855 ∪ ciun 4939 ↦ cmpt 5171 ^◡ccnv 5635 dom cdm 5636 ran crn 5637 “ cima 5639 ⟶wf 6502 –onto→wfo 6504 (class class class)co 7381 Fincfn 8912 Topctop 22922 Cn ccn 23253 Compccmp 23415

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1805 ax-4 1819 ax-5 1920 ax-6 1977 ax-7 2018 ax-8 2134 ax-9 2142 ax-10 2165 ax-11 2181 ax-12 2202 ax-ext 2724 ax-sep 5236 ax-nul 5246 ax-pow 5312 ax-pr 5380 ax-un 7703

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1553 df-fal 1563 df-ex 1790 df-nf 1794 df-sb 2081 df-mo 2556 df-eu 2586 df-clab 2731 df-cleq 2744 df-clel 2827 df-nfc 2901 df-ne 2948 df-ral 3067 df-rex 3077 df-reu 3358 df-rab 3405 df-v 3446 df-sbc 3736 df-csb 3844 df-dif 3898 df-un 3900 df-in 3902 df-ss 3912 df-pss 3915 df-nul 4277 df-if 4471 df-pw 4547 df-sn 4573 df-pr 4575 df-op 4579 df-uni 4856 df-iun 4941 df-br 5091 df-opab 5153 df-mpt 5172 df-tr 5198 df-id 5531 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5589 df-we 5591 df-xp 5642 df-rel 5643 df-cnv 5644 df-co 5645 df-dm 5646 df-rn 5647 df-res 5648 df-ima 5649 df-ord 6334 df-on 6335 df-lim 6336 df-suc 6337 df-iota 6462 df-fun 6508 df-fn 6509 df-f 6510 df-f1 6511 df-fo 6512 df-f1o 6513 df-fv 6514 df-ov 7384 df-oprab 7385 df-mpo 7386 df-om 7832 df-1st 7955 df-2nd 7956 df-1o 8421 df-map 8794 df-en 8913 df-dom 8914 df-fin 8916 df-top 22923 df-topon 22940 df-cn 23256 df-cmp 23416

This theorem is referenced by: rncmp 23425 txcmpb 23673 qtopcmp 23737 cmphmph 23817

W3C validator