Theorem qtopcmap 23906

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 23902	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) → 𝐾 ⊆ (𝐽 qTop 𝐹))
5	1, 2, 3, 4	syl3anc 1398	. 2 ⊢ (𝜑 → 𝐾 ⊆ (𝐽 qTop 𝐹))
6		cntop1 23427	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
7	1, 6	syl 18	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top)
8		toptopon2 23105	. . . . . . . . 9 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
9	7, 8	sylib 221	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘∪ 𝐽))
10		cnf2 23436	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:∪ 𝐽⟶𝑌)
11	9, 2, 1, 10	syl3anc 1398	. . . . . . 7 ⊢ (𝜑 → 𝐹:∪ 𝐽⟶𝑌)
12	11	ffnd 6713	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn ∪ 𝐽)
13		df-fo 6549	. . . . . 6 ⊢ (𝐹:∪ 𝐽–onto→𝑌 ↔ (𝐹 Fn ∪ 𝐽 ∧ ran 𝐹 = 𝑌))
14	12, 3, 13	sylanbrc 595	. . . . 5 ⊢ (𝜑 → 𝐹:∪ 𝐽–onto→𝑌)
15		eqid 2766	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
16	15	elqtop2 23888	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐹:∪ 𝐽–onto→𝑌) → (𝑦 ∈ (𝐽 qTop 𝐹) ↔ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)))
17	7, 14, 16	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝐽 qTop 𝐹) ↔ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)))
18	14	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → 𝐹:∪ 𝐽–onto→𝑌)
19		difss 4093	. . . . . . . . 9 ⊢ (𝑌 ∖ 𝑦) ⊆ 𝑌
20		foimacnv 6845	. . . . . . . . 9 ⊢ ((𝐹:∪ 𝐽–onto→𝑌 ∧ (𝑌 ∖ 𝑦) ⊆ 𝑌) → (𝐹 “ (◡𝐹 “ (𝑌 ∖ 𝑦))) = (𝑌 ∖ 𝑦))
21	18, 19, 20	sylancl 598	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝐹 “ (◡𝐹 “ (𝑌 ∖ 𝑦))) = (𝑌 ∖ 𝑦))
22	2	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → 𝐾 ∈ (TopOn‘𝑌))
23		toponuni 23101	. . . . . . . . . 10 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
24	22, 23	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → 𝑌 = ∪ 𝐾)
25	24	difeq1d 4083	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝑌 ∖ 𝑦) = (∪ 𝐾 ∖ 𝑦))
26	21, 25	eqtrd 2801	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝐹 “ (◡𝐹 “ (𝑌 ∖ 𝑦))) = (∪ 𝐾 ∖ 𝑦))
27		imaeq2 6063	. . . . . . . . 9 ⊢ (𝑥 = (◡𝐹 “ (𝑌 ∖ 𝑦)) → (𝐹 “ 𝑥) = (𝐹 “ (◡𝐹 “ (𝑌 ∖ 𝑦))))
28	27	eleq1d 2851	. . . . . . . 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 3160	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ (Clsd‘𝐽)(𝐹 “ 𝑥) ∈ (Clsd‘𝐾))
31	30	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → ∀𝑥 ∈ (Clsd‘𝐽)(𝐹 “ 𝑥) ∈ (Clsd‘𝐾))
32		fofun 6800	. . . . . . . . . . 11 ⊢ (𝐹:∪ 𝐽–onto→𝑌 → Fun 𝐹)
33		funcnvcnv 6610	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → Fun ◡◡𝐹)
34		imadif 6627	. . . . . . . . . . 11 ⊢ (Fun ◡◡𝐹 → (◡𝐹 “ (𝑌 ∖ 𝑦)) = ((◡𝐹 “ 𝑌) ∖ (◡𝐹 “ 𝑦)))
35	18, 32, 33, 34	4syl 20	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (◡𝐹 “ (𝑌 ∖ 𝑦)) = ((◡𝐹 “ 𝑌) ∖ (◡𝐹 “ 𝑦)))
36	11	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → 𝐹:∪ 𝐽⟶𝑌)
37		fimacnv 6735	. . . . . . . . . . . 12 ⊢ (𝐹:∪ 𝐽⟶𝑌 → (◡𝐹 “ 𝑌) = ∪ 𝐽)
38	36, 37	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (◡𝐹 “ 𝑌) = ∪ 𝐽)
39	38	difeq1d 4083	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → ((◡𝐹 “ 𝑌) ∖ (◡𝐹 “ 𝑦)) = (∪ 𝐽 ∖ (◡𝐹 “ 𝑦)))
40	35, 39	eqtrd 2801	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (◡𝐹 “ (𝑌 ∖ 𝑦)) = (∪ 𝐽 ∖ (◡𝐹 “ 𝑦)))
41	7	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → 𝐽 ∈ Top)
42		simprr 785	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (◡𝐹 “ 𝑦) ∈ 𝐽)
43	15	opncld 23220	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ (◡𝐹 “ 𝑦) ∈ 𝐽) → (∪ 𝐽 ∖ (◡𝐹 “ 𝑦)) ∈ (Clsd‘𝐽))
44	41, 42, 43	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (∪ 𝐽 ∖ (◡𝐹 “ 𝑦)) ∈ (Clsd‘𝐽))
45	40, 44	eqeltrd 2866	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (◡𝐹 “ (𝑌 ∖ 𝑦)) ∈ (Clsd‘𝐽))
46	28, 31, 45	rspcdva 3585	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝐹 “ (◡𝐹 “ (𝑌 ∖ 𝑦))) ∈ (Clsd‘𝐾))
47	26, 46	eqeltrrd 2867	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (∪ 𝐾 ∖ 𝑦) ∈ (Clsd‘𝐾))
48		topontop 23100	. . . . . . . 8 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
49	22, 48	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → 𝐾 ∈ Top)
50		simprl 783	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → 𝑦 ⊆ 𝑌)
51	50, 24	sseqtrd 3976	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → 𝑦 ⊆ ∪ 𝐾)
52		eqid 2766	. . . . . . . 8 ⊢ ∪ 𝐾 = ∪ 𝐾
53	52	isopn2 23219	. . . . . . 7 ⊢ ((𝐾 ∈ Top ∧ 𝑦 ⊆ ∪ 𝐾) → (𝑦 ∈ 𝐾 ↔ (∪ 𝐾 ∖ 𝑦) ∈ (Clsd‘𝐾)))
54	49, 51, 53	syl2anc 596	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝑦 ∈ 𝐾 ↔ (∪ 𝐾 ∖ 𝑦) ∈ (Clsd‘𝐾)))
55	47, 54	mpbird 260	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → 𝑦 ∈ 𝐾)
56	55	ex 418	. . . 4 ⊢ (𝜑 → ((𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽) → 𝑦 ∈ 𝐾))
57	17, 56	sylbid 243	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐽 qTop 𝐹) → 𝑦 ∈ 𝐾))
58	57	ssrdv 3946	. 2 ⊢ (𝜑 → (𝐽 qTop 𝐹) ⊆ 𝐾)
59	5, 58	eqssd 3957	1 ⊢ (𝜑 → 𝐾 = (𝐽 qTop 𝐹))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3082   ∖ cdif 3905   ⊆ wss 3908  ∪ cuni 4877  ^◡ccnv 5665  ran crn 5667   “ cima 5669  Fun wfun 6537   Fn wfn 6538  ⟶wf 6539  –onto→wfo 6541  ‘cfv 6543  (class class class)co 7423   qTop cqtop 17582  Topctop 23080  TopOnctopon 23097  Clsdccld 23203   Cn ccn 23411

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-map 8835  df-qtop 17586  df-top 23081  df-topon 23098  df-cld 23206  df-cn 23414

This theorem is used by: (None)