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Theorem cncmp 23671

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 23520	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
2	1	3ad2ant3 1153	. 2 ⊢ ((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
3		elpwi 4563	. . . 4 ⊢ (𝑢 ∈ 𝒫 𝐾 → 𝑢 ⊆ 𝐾)
4		simpl1 1210	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → 𝐽 ∈ Comp)
5		simpl3 1212	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → 𝐹 ∈ (𝐽 Cn 𝐾))
6		simprl 783	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → 𝑢 ⊆ 𝐾)
7	6	sselda 3930	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑦 ∈ 𝑢) → 𝑦 ∈ 𝐾)
8		cnima 23544	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 ∈ 𝐾) → (^◡𝐹 “ 𝑦) ∈ 𝐽)
9	5, 7, 8	syl2an2r 698	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑦 ∈ 𝑢) → (^◡𝐹 “ 𝑦) ∈ 𝐽)
10	9	fmpttd 7103	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)):𝑢⟶𝐽)
11	10	frnd 6706	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ⊆ 𝐽)
12		simprr 785	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → 𝑌 = ∪ 𝑢)
13	12	imaeq2d 6050	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → (^◡𝐹 “ 𝑌) = (^◡𝐹 “ ∪ 𝑢))
14		eqid 2760	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
15		cncmp.2	. . . . . . . . . . 11 ⊢ 𝑌 = ∪ 𝐾
16	14, 15	cnf 23525	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶𝑌)
17	5, 16	syl 18	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → 𝐹:∪ 𝐽⟶𝑌)
18		fimacnv 6720	. . . . . . . . 9 ⊢ (𝐹:∪ 𝐽⟶𝑌 → (^◡𝐹 “ 𝑌) = ∪ 𝐽)
19	17, 18	syl 18	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → (^◡𝐹 “ 𝑌) = ∪ 𝐽)
20	9	ralrimiva 3154	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → ∀𝑦 ∈ 𝑢 (^◡𝐹 “ 𝑦) ∈ 𝐽)
21		dfiun2g 4987	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑢 (^◡𝐹 “ 𝑦) ∈ 𝐽 → ∪ 𝑦 ∈ 𝑢 (^◡𝐹 “ 𝑦) = ∪ {𝑥 ∣ ∃𝑦 ∈ 𝑢 𝑥 = (^◡𝐹 “ 𝑦)})
22	20, 21	syl 18	. . . . . . . . 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 7238	. . . . . . . . 9 ⊢ (^◡𝐹 “ ∪ 𝑢) = ∪ 𝑦 ∈ 𝑢 (^◡𝐹 “ 𝑦)
24		eqid 2760	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) = (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦))
25	24	rnmpt 5935	. . . . . . . . . 10 ⊢ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) = {𝑥 ∣ ∃𝑦 ∈ 𝑢 𝑥 = (^◡𝐹 “ 𝑦)}
26	25	unieqi 4878	. . . . . . . . 9 ⊢ ∪ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) = ∪ {𝑥 ∣ ∃𝑦 ∈ 𝑢 𝑥 = (^◡𝐹 “ 𝑦)}
27	22, 23, 26	3eqtr4g 2820	. . . . . . . 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 2803	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → ∪ 𝐽 = ∪ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)))
29	14	cmpcov 23668	. . . . . . 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 1398	. . . . . 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 9321	. . . . . . . 8 ⊢ (𝑠 ∈ (𝒫 ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ∩ Fin) ↔ (𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (^◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin))
32		simprll 791	. . . . . . . . . . . . . . 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 3930	. . . . . . . . . . . . . 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 1232	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑦 ∈ 𝑢) → 𝐹:𝑋–onto→𝑌)
35		elssuni 4898	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ 𝐾 → 𝑦 ⊆ ∪ 𝐾)
36	35, 15	sseqtrrdi 3971	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ 𝐾 → 𝑦 ⊆ 𝑌)
37	7, 36	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑦 ∈ 𝑢) → 𝑦 ⊆ 𝑌)
38		foimacnv 6830	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑦 ⊆ 𝑌) → (𝐹 “ (^◡𝐹 “ 𝑦)) = 𝑦)
39	34, 37, 38	syl2anc 596	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑦 ∈ 𝑢) → (𝐹 “ (^◡𝐹 “ 𝑦)) = 𝑦)
40		simpr 490	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑦 ∈ 𝑢) → 𝑦 ∈ 𝑢)
41	39, 40	eqeltrd 2860	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑦 ∈ 𝑢) → (𝐹 “ (^◡𝐹 “ 𝑦)) ∈ 𝑢)
42	41	ralrimiva 3154	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → ∀𝑦 ∈ 𝑢 (𝐹 “ (^◡𝐹 “ 𝑦)) ∈ 𝑢)
43		imaeq2 6046	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 = (^◡𝐹 “ 𝑦) → (𝐹 “ 𝑐) = (𝐹 “ (^◡𝐹 “ 𝑦)))
44	43	eleq1d 2845	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 = (^◡𝐹 “ 𝑦) → ((𝐹 “ 𝑐) ∈ 𝑢 ↔ (𝐹 “ (^◡𝐹 “ 𝑦)) ∈ 𝑢))
45	24, 44	ralrnmptw 7082	. . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . 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 3254	. . . . . . . . . . . . . 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 603	. . . . . . . . . . . . 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 7103	. . . . . . . . . . . 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 6706	. . . . . . . . . . 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 792	. . . . . . . . . . . 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 2760	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) = (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐))
55	54	rnmpt 5935	. . . . . . . . . . . . 13 ⊢ ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) = {𝑑 ∣ ∃𝑐 ∈ 𝑠 𝑑 = (𝐹 “ 𝑐)}
56		abrexfi 9319	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ Fin → {𝑑 ∣ ∃𝑐 ∈ 𝑠 𝑑 = (𝐹 “ 𝑐)} ∈ Fin)
57	55, 56	eqeltrid 2864	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ Fin → ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) ∈ Fin)
58	53, 57	syl 18	. . . . . . . . . . 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 9321	. . . . . . . . . . 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 595	. . . . . . . . . 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 486	. . . . . . . . . . . . . 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 6708	. . . . . . . . . . . . 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 1232	. . . . . . . . . . . . . 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 6784	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌)
65		fdm 6707	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋)
66	63, 64, 65	3syl 19	. . . . . . . . . . . . 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 785	. . . . . . . . . . . . 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 2803	. . . . . . . . . . . 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 6050	. . . . . . . . . . 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 6789	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋–onto→𝑌 → (𝐹 “ 𝑋) = 𝑌)
71	63, 70	syl 18	. . . . . . . . . . 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 3154	. . . . . . . . . . . . 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 4987	. . . . . . . . . . . . 13 ⊢ (∀𝑐 ∈ 𝑠 (𝐹 “ 𝑐) ∈ 𝑢 → ∪ 𝑐 ∈ 𝑠 (𝐹 “ 𝑐) = ∪ {𝑑 ∣ ∃𝑐 ∈ 𝑠 𝑑 = (𝐹 “ 𝑐)})
74	72, 73	syl 18	. . . . . . . . . . . 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 7238	. . . . . . . . . . . 12 ⊢ (𝐹 “ ∪ 𝑠) = ∪ 𝑐 ∈ 𝑠 (𝐹 “ 𝑐)
76	55	unieqi 4878	. . . . . . . . . . . 12 ⊢ ∪ ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) = ∪ {𝑑 ∣ ∃𝑐 ∈ 𝑠 𝑑 = (𝐹 “ 𝑐)}
77	74, 75, 76	3eqtr4g 2820	. . . . . . . . . . 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 2803	. . . . . . . . . 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 4877	. . . . . . . . . . 11 ⊢ (𝑣 = ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) → ∪ 𝑣 = ∪ ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)))
80	79	rspceeqv 3598	. . . . . . . . . 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 596	. . . . . . . . 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 462	. . . . . . . 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 606	. . . . . . 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 3163	. . . . . 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 16	. . . . 5 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = ∪ 𝑣)
86	85	expr 462	. . . 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 605	. . 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 3154	. 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 23667	. 2 ⊢ (𝐾 ∈ Comp ↔ (𝐾 ∈ Top ∧ ∀𝑢 ∈ 𝒫 𝐾(𝑌 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = ∪ 𝑣)))
90	2, 88, 89	sylanbrc 595	1 ⊢ ((𝐽 ∈ Comp ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Comp)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 {cab 2738 ∀wral 3076 ∃wrex 3086 ∩ cin 3897 ⊆ wss 3898 𝒫 cpw 4556 ∪ cuni 4866 ∪ ciun 4950 ↦ cmpt 5185 ^◡ccnv 5646 dom cdm 5647 ran crn 5648 “ cima 5650 ⟶wf 6523 –onto→wfo 6525 (class class class)co 7408 Fincfn 8951 Topctop 23172 Cn ccn 23503 Compccmp 23665

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7734

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-ord 6354 df-on 6355 df-lim 6356 df-suc 6357 df-iota 6483 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7861 df-1st 7984 df-2nd 7985 df-1o 8454 df-map 8827 df-en 8952 df-dom 8953 df-fin 8955 df-top 23173 df-topon 23190 df-cn 23506 df-cmp 23666

This theorem is used by: rncmp 23675 txcmpb 23924 qtopcmp 23988 cmphmph 24068

W3C validator