Theorem qtopcmap 23748

Description: If 𝐹 is a surjective continuous closed map, then it is a quotient map. (A closed map is a function that maps closed sets to closed sets.) (Contributed by Mario Carneiro, 24-Mar-2015.)

Hypotheses

Ref	Expression
qtopomap.4	⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
qtopomap.5	⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
qtopomap.6	⊢ (𝜑 → ran 𝐹 = 𝑌)
qtopcmap.7	⊢ ((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝐹 “ 𝑥) ∈ (Clsd‘𝐾))

Assertion

Ref	Expression
qtopcmap	⊢ (𝜑 → 𝐾 = (𝐽 qTop 𝐹))

Distinct variable groups:   𝑥,𝐹   𝑥,𝐽   𝑥,𝐾   𝜑,𝑥   𝑥,𝑌

Proof of Theorem qtopcmap

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		qtopomap.5	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
2		qtopomap.4	. . 3 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
3		qtopomap.6	. . 3 ⊢ (𝜑 → ran 𝐹 = 𝑌)
4		qtopss 23744	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) → 𝐾 ⊆ (𝐽 qTop 𝐹))
5	1, 2, 3, 4	syl3anc 1371	. 2 ⊢ (𝜑 → 𝐾 ⊆ (𝐽 qTop 𝐹))
6		cntop1 23269	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
7	1, 6	syl 17	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top)
8		toptopon2 22945	. . . . . . . . 9 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
9	7, 8	sylib 218	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘∪ 𝐽))
10		cnf2 23278	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:∪ 𝐽⟶𝑌)
11	9, 2, 1, 10	syl3anc 1371	. . . . . . 7 ⊢ (𝜑 → 𝐹:∪ 𝐽⟶𝑌)
12	11	ffnd 6748	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn ∪ 𝐽)
13		df-fo 6579	. . . . . 6 ⊢ (𝐹:∪ 𝐽–onto→𝑌 ↔ (𝐹 Fn ∪ 𝐽 ∧ ran 𝐹 = 𝑌))
14	12, 3, 13	sylanbrc 582	. . . . 5 ⊢ (𝜑 → 𝐹:∪ 𝐽–onto→𝑌)
15		eqid 2740	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
16	15	elqtop2 23730	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐹:∪ 𝐽–onto→𝑌) → (𝑦 ∈ (𝐽 qTop 𝐹) ↔ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)))
17	7, 14, 16	syl2anc 583	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝐽 qTop 𝐹) ↔ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)))
18	14	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → 𝐹:∪ 𝐽–onto→𝑌)
19		difss 4159	. . . . . . . . 9 ⊢ (𝑌 ∖ 𝑦) ⊆ 𝑌
20		foimacnv 6879	. . . . . . . . 9 ⊢ ((𝐹:∪ 𝐽–onto→𝑌 ∧ (𝑌 ∖ 𝑦) ⊆ 𝑌) → (𝐹 “ (◡𝐹 “ (𝑌 ∖ 𝑦))) = (𝑌 ∖ 𝑦))
21	18, 19, 20	sylancl 585	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝐹 “ (◡𝐹 “ (𝑌 ∖ 𝑦))) = (𝑌 ∖ 𝑦))
22	2	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → 𝐾 ∈ (TopOn‘𝑌))
23		toponuni 22941	. . . . . . . . . 10 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
24	22, 23	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → 𝑌 = ∪ 𝐾)
25	24	difeq1d 4148	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝑌 ∖ 𝑦) = (∪ 𝐾 ∖ 𝑦))
26	21, 25	eqtrd 2780	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝐹 “ (◡𝐹 “ (𝑌 ∖ 𝑦))) = (∪ 𝐾 ∖ 𝑦))
27		imaeq2 6085	. . . . . . . . 9 ⊢ (𝑥 = (◡𝐹 “ (𝑌 ∖ 𝑦)) → (𝐹 “ 𝑥) = (𝐹 “ (◡𝐹 “ (𝑌 ∖ 𝑦))))
28	27	eleq1d 2829	. . . . . . . 8 ⊢ (𝑥 = (◡𝐹 “ (𝑌 ∖ 𝑦)) → ((𝐹 “ 𝑥) ∈ (Clsd‘𝐾) ↔ (𝐹 “ (◡𝐹 “ (𝑌 ∖ 𝑦))) ∈ (Clsd‘𝐾)))
29		qtopcmap.7	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝐹 “ 𝑥) ∈ (Clsd‘𝐾))
30	29	ralrimiva 3152	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ (Clsd‘𝐽)(𝐹 “ 𝑥) ∈ (Clsd‘𝐾))
31	30	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → ∀𝑥 ∈ (Clsd‘𝐽)(𝐹 “ 𝑥) ∈ (Clsd‘𝐾))
32		fofun 6835	. . . . . . . . . . 11 ⊢ (𝐹:∪ 𝐽–onto→𝑌 → Fun 𝐹)
33		funcnvcnv 6645	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → Fun ◡◡𝐹)
34		imadif 6662	. . . . . . . . . . 11 ⊢ (Fun ◡◡𝐹 → (◡𝐹 “ (𝑌 ∖ 𝑦)) = ((◡𝐹 “ 𝑌) ∖ (◡𝐹 “ 𝑦)))
35	18, 32, 33, 34	4syl 19	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (◡𝐹 “ (𝑌 ∖ 𝑦)) = ((◡𝐹 “ 𝑌) ∖ (◡𝐹 “ 𝑦)))
36	11	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → 𝐹:∪ 𝐽⟶𝑌)
37		fimacnv 6769	. . . . . . . . . . . 12 ⊢ (𝐹:∪ 𝐽⟶𝑌 → (◡𝐹 “ 𝑌) = ∪ 𝐽)
38	36, 37	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (◡𝐹 “ 𝑌) = ∪ 𝐽)
39	38	difeq1d 4148	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → ((◡𝐹 “ 𝑌) ∖ (◡𝐹 “ 𝑦)) = (∪ 𝐽 ∖ (◡𝐹 “ 𝑦)))
40	35, 39	eqtrd 2780	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (◡𝐹 “ (𝑌 ∖ 𝑦)) = (∪ 𝐽 ∖ (◡𝐹 “ 𝑦)))
41	7	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → 𝐽 ∈ Top)
42		simprr 772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (◡𝐹 “ 𝑦) ∈ 𝐽)
43	15	opncld 23062	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ (◡𝐹 “ 𝑦) ∈ 𝐽) → (∪ 𝐽 ∖ (◡𝐹 “ 𝑦)) ∈ (Clsd‘𝐽))
44	41, 42, 43	syl2anc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (∪ 𝐽 ∖ (◡𝐹 “ 𝑦)) ∈ (Clsd‘𝐽))
45	40, 44	eqeltrd 2844	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (◡𝐹 “ (𝑌 ∖ 𝑦)) ∈ (Clsd‘𝐽))
46	28, 31, 45	rspcdva 3636	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝐹 “ (◡𝐹 “ (𝑌 ∖ 𝑦))) ∈ (Clsd‘𝐾))
47	26, 46	eqeltrrd 2845	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (∪ 𝐾 ∖ 𝑦) ∈ (Clsd‘𝐾))
48		topontop 22940	. . . . . . . 8 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
49	22, 48	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → 𝐾 ∈ Top)
50		simprl 770	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → 𝑦 ⊆ 𝑌)
51	50, 24	sseqtrd 4049	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → 𝑦 ⊆ ∪ 𝐾)
52		eqid 2740	. . . . . . . 8 ⊢ ∪ 𝐾 = ∪ 𝐾
53	52	isopn2 23061	. . . . . . 7 ⊢ ((𝐾 ∈ Top ∧ 𝑦 ⊆ ∪ 𝐾) → (𝑦 ∈ 𝐾 ↔ (∪ 𝐾 ∖ 𝑦) ∈ (Clsd‘𝐾)))
54	49, 51, 53	syl2anc 583	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝑦 ∈ 𝐾 ↔ (∪ 𝐾 ∖ 𝑦) ∈ (Clsd‘𝐾)))
55	47, 54	mpbird 257	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → 𝑦 ∈ 𝐾)
56	55	ex 412	. . . 4 ⊢ (𝜑 → ((𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽) → 𝑦 ∈ 𝐾))
57	17, 56	sylbid 240	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐽 qTop 𝐹) → 𝑦 ∈ 𝐾))
58	57	ssrdv 4014	. 2 ⊢ (𝜑 → (𝐽 qTop 𝐹) ⊆ 𝐾)
59	5, 58	eqssd 4026	1 ⊢ (𝜑 → 𝐾 = (𝐽 qTop 𝐹))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ∖ cdif 3973   ⊆ wss 3976  ∪ cuni 4931  ^◡ccnv 5699  ran crn 5701   “ cima 5703  Fun wfun 6567   Fn wfn 6568  ⟶wf 6569  –onto→wfo 6571  ‘cfv 6573  (class class class)co 7448   qTop cqtop 17563  Topctop 22920  TopOnctopon 22937  Clsdccld 23045   Cn ccn 23253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-map 8886  df-qtop 17567  df-top 22921  df-topon 22938  df-cld 23048  df-cn 23256

This theorem is referenced by: (None)