Theorem cncmp 23740

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⟶onto𝑌 ∧ 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 23589	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
2	1	3ad2ant3 1153	. 2 ⊢ ((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
3		elpwi 4567	. . . 4 ⊢ (𝑢 ∈ 𝒫 𝐾 → 𝑢 ⊆ 𝐾)
4		simpl1 1210	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → 𝐽 ∈ Comp)
5		simpl3 1212	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → 𝐹 ∈ (𝐽 Cn 𝐾))
6		simprl 783	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → 𝑢 ⊆ 𝐾)
7	6	sselda 3913	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑦 ∈ 𝑢) → 𝑦 ∈ 𝐾)
8		cnima 23613	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 ∈ 𝐾) → (◡𝐹 “ 𝑦) ∈ 𝐽)
9	5, 7, 8	syl2an2r 698	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑦 ∈ 𝑢) → (◡𝐹 “ 𝑦) ∈ 𝐽)
10	9	fmpttd 7139	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)):𝑢⟶𝐽)
11	10	frnd 6740	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ⊆ 𝐽)
12		simprr 785	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → 𝑌 = ∪ 𝑢)
13	12	imaeq2d 6064	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → (◡𝐹 “ 𝑌) = (◡𝐹 “ ∪ 𝑢))
14		eqid 2763	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
15		cncmp.2	. . . . . . . . . . 11 ⊢ 𝑌 = ∪ 𝐾
16	14, 15	cnf 23594	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶𝑌)
17	5, 16	syl 18	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → 𝐹:∪ 𝐽⟶𝑌)
18		fimacnv 6754	. . . . . . . . 9 ⊢ (𝐹:∪ 𝐽⟶𝑌 → (◡𝐹 “ 𝑌) = ∪ 𝐽)
19	17, 18	syl 18	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → (◡𝐹 “ 𝑌) = ∪ 𝐽)
20	9	ralrimiva 3158	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → ∀𝑦 ∈ 𝑢 (◡𝐹 “ 𝑦) ∈ 𝐽)
21		dfiun2g 4991	. . . . . . . . . 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⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → ∪ 𝑦 ∈ 𝑢 (◡𝐹 “ 𝑦) = ∪ {𝑥 ∣ ∃𝑦 ∈ 𝑢 𝑥 = (◡𝐹 “ 𝑦)})
23		imauni 7273	. . . . . . . . 9 ⊢ (◡𝐹 “ ∪ 𝑢) = ∪ 𝑦 ∈ 𝑢 (◡𝐹 “ 𝑦)
24		eqid 2763	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) = (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦))
25	24	rnmpt 5946	. . . . . . . . . 10 ⊢ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) = {𝑥 ∣ ∃𝑦 ∈ 𝑢 𝑥 = (◡𝐹 “ 𝑦)}
26	25	unieqi 4882	. . . . . . . . 9 ⊢ ∪ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) = ∪ {𝑥 ∣ ∃𝑦 ∈ 𝑢 𝑥 = (◡𝐹 “ 𝑦)}
27	22, 23, 26	3eqtr4g 2823	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → (◡𝐹 “ ∪ 𝑢) = ∪ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)))
28	13, 19, 27	3eqtr3d 2806	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → ∪ 𝐽 = ∪ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)))
29	14	cmpcov 23737	. . . . . . 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 1399	. . . . . 6 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → ∃𝑠 ∈ (𝒫 ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∩ Fin)∪ 𝐽 = ∪ 𝑠)
31		elfpw 9371	. . . . . . . 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⟶onto𝑌 ∧ 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 3913	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 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⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑦 ∈ 𝑢) → 𝐹:𝑋⟶onto𝑌)
35		elssuni 4902	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ 𝐾 → 𝑦 ⊆ ∪ 𝐾)
36	35, 15	sseqtrrdi 3954	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ 𝐾 → 𝑦 ⊆ 𝑌)
37	7, 36	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑦 ∈ 𝑢) → 𝑦 ⊆ 𝑌)
38		foimacnv 6864	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹:𝑋⟶onto𝑌 ∧ 𝑦 ⊆ 𝑌) → (𝐹 “ (◡𝐹 “ 𝑦)) = 𝑦)
39	34, 37, 38	syl2anc 596	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑦 ∈ 𝑢) → (𝐹 “ (◡𝐹 “ 𝑦)) = 𝑦)
40		simpr 490	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑦 ∈ 𝑢) → 𝑦 ∈ 𝑢)
41	39, 40	eqeltrd 2864	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ 𝑦 ∈ 𝑢) → (𝐹 “ (◡𝐹 “ 𝑦)) ∈ 𝑢)
42	41	ralrimiva 3158	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → ∀𝑦 ∈ 𝑢 (𝐹 “ (◡𝐹 “ 𝑦)) ∈ 𝑢)
43		imaeq2 6060	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 = (◡𝐹 “ 𝑦) → (𝐹 “ 𝑐) = (𝐹 “ (◡𝐹 “ 𝑦)))
44	43	eleq1d 2849	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 = (◡𝐹 “ 𝑦) → ((𝐹 “ 𝑐) ∈ 𝑢 ↔ (𝐹 “ (◡𝐹 “ 𝑦)) ∈ 𝑢))
45	24, 44	ralrnmptw 7118	. . . . . . . . . . . . . . . . . 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⟶onto𝑌 ∧ 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⟶onto𝑌 ∧ 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⟶onto𝑌 ∧ 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 3258	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 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⟶onto𝑌 ∧ 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 7139	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 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 6740	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 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⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑠)) → 𝑠 ∈ Fin)
54		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) = (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐))
55	54	rnmpt 5946	. . . . . . . . . . . . 13 ⊢ ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) = {𝑑 ∣ ∃𝑐 ∈ 𝑠 𝑑 = (𝐹 “ 𝑐)}
56		abrexfi 9369	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ Fin → {𝑑 ∣ ∃𝑐 ∈ 𝑠 𝑑 = (𝐹 “ 𝑐)} ∈ Fin)
57	55, 56	eqeltrid 2868	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ Fin → ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) ∈ Fin)
58	53, 57	syl 18	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 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 9371	. . . . . . . . . . 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⟶onto𝑌 ∧ 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⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑠)) → 𝐹:∪ 𝐽⟶𝑌)
62	61	fdmd 6742	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 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⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑠)) → 𝐹:𝑋⟶onto𝑌)
64		fof 6818	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝑋⟶onto𝑌 → 𝐹:𝑋⟶𝑌)
65		fdm 6741	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋)
66	63, 64, 65	3syl 19	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 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⟶onto𝑌 ∧ 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 2806	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑠)) → 𝑋 = ∪ 𝑠)
69	68	imaeq2d 6064	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑠)) → (𝐹 “ 𝑋) = (𝐹 “ ∪ 𝑠))
70		foima 6823	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋⟶onto𝑌 → (𝐹 “ 𝑋) = 𝑌)
71	63, 70	syl 18	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦 ∈ 𝑢 ↦ (◡𝐹 “ 𝑦)) ∧ 𝑠 ∈ Fin) ∧ ∪ 𝐽 = ∪ 𝑠)) → (𝐹 “ 𝑋) = 𝑌)
72	50	ralrimiva 3158	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 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 4991	. . . . . . . . . . . . 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⟶onto𝑌 ∧ 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 7273	. . . . . . . . . . . 12 ⊢ (𝐹 “ ∪ 𝑠) = ∪ 𝑐 ∈ 𝑠 (𝐹 “ 𝑐)
76	55	unieqi 4882	. . . . . . . . . . . 12 ⊢ ∪ ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) = ∪ {𝑑 ∣ ∃𝑐 ∈ 𝑠 𝑑 = (𝐹 “ 𝑐)}
77	74, 75, 76	3eqtr4g 2823	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 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 2806	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 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 4881	. . . . . . . . . . 11 ⊢ (𝑣 = ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)) → ∪ 𝑣 = ∪ ran (𝑐 ∈ 𝑠 ↦ (𝐹 “ 𝑐)))
80	79	rspceeqv 3602	. . . . . . . . . 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⟶onto𝑌 ∧ 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⟶onto𝑌 ∧ 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⟶onto𝑌 ∧ 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 3167	. . . . . 6 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 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⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 ⊆ 𝐾 ∧ 𝑌 = ∪ 𝑢)) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = ∪ 𝑣)
86	85	expr 462	. . . 4 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑢 ⊆ 𝐾) → (𝑌 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = ∪ 𝑣))
87	3, 86	sylan2 605	. . 3 ⊢ (((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑢 ∈ 𝒫 𝐾) → (𝑌 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = ∪ 𝑣))
88	87	ralrimiva 3158	. 2 ⊢ ((𝐽 ∈ Comp ∧ 𝐹:𝑋⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑢 ∈ 𝒫 𝐾(𝑌 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = ∪ 𝑣))
89	15	iscmp 23736	. 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⟶onto𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Comp)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   ∩ cin 4092  𝒫 cpw 4560  ∪ cuni 4870  ∪ ciun 4954   ↦ cmpt 5189  ^◡ccnv 5654  dom cdm 5655  ran crn 5656   “ cima 5658  ⟶wf 6554  ⟶ontowfo 6556  (class class class)co 7443  Fincfn 9001  Topctop 23241   Cn ccn 23572  Compccmp 23734

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-1o 8504  df-map 8877  df-en 9002  df-dom 9003  df-fin 9005  df-top 23242  df-topon 23259  df-cn 23575  df-cmp 23735

This theorem is used by:  rncmp  23744  txcmpb  23993  qtopcmp  24057  cmphmph  24137