Theorem qtopcmap 23757

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 23753	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) → 𝐾 ⊆ (𝐽 qTop 𝐹))
5	1, 2, 3, 4	syl3anc 1389	. 2 ⊢ (𝜑 → 𝐾 ⊆ (𝐽 qTop 𝐹))
6		cntop1 23278	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
7	1, 6	syl 17	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top)
8		toptopon2 22956	. . . . . . . . 9 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
9	7, 8	sylib 220	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘∪ 𝐽))
10		cnf2 23287	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:∪ 𝐽⟶𝑌)
11	9, 2, 1, 10	syl3anc 1389	. . . . . . 7 ⊢ (𝜑 → 𝐹:∪ 𝐽⟶𝑌)
12	11	ffnd 6686	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn ∪ 𝐽)
13		df-fo 6521	. . . . . 6 ⊢ (𝐹:∪ 𝐽–onto→𝑌 ↔ (𝐹 Fn ∪ 𝐽 ∧ ran 𝐹 = 𝑌))
14	12, 3, 13	sylanbrc 592	. . . . 5 ⊢ (𝜑 → 𝐹:∪ 𝐽–onto→𝑌)
15		eqid 2761	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
16	15	elqtop2 23739	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐹:∪ 𝐽–onto→𝑌) → (𝑦 ∈ (𝐽 qTop 𝐹) ↔ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)))
17	7, 14, 16	syl2anc 593	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝐽 qTop 𝐹) ↔ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)))
18	14	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → 𝐹:∪ 𝐽–onto→𝑌)
19		difss 4089	. . . . . . . . 9 ⊢ (𝑌 ∖ 𝑦) ⊆ 𝑌
20		foimacnv 6818	. . . . . . . . 9 ⊢ ((𝐹:∪ 𝐽–onto→𝑌 ∧ (𝑌 ∖ 𝑦) ⊆ 𝑌) → (𝐹 “ (◡𝐹 “ (𝑌 ∖ 𝑦))) = (𝑌 ∖ 𝑦))
21	18, 19, 20	sylancl 595	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝐹 “ (◡𝐹 “ (𝑌 ∖ 𝑦))) = (𝑌 ∖ 𝑦))
22	2	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → 𝐾 ∈ (TopOn‘𝑌))
23		toponuni 22952	. . . . . . . . . 10 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
24	22, 23	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → 𝑌 = ∪ 𝐾)
25	24	difeq1d 4079	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝑌 ∖ 𝑦) = (∪ 𝐾 ∖ 𝑦))
26	21, 25	eqtrd 2796	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝐹 “ (◡𝐹 “ (𝑌 ∖ 𝑦))) = (∪ 𝐾 ∖ 𝑦))
27		imaeq2 6040	. . . . . . . . 9 ⊢ (𝑥 = (◡𝐹 “ (𝑌 ∖ 𝑦)) → (𝐹 “ 𝑥) = (𝐹 “ (◡𝐹 “ (𝑌 ∖ 𝑦))))
28	27	eleq1d 2846	. . . . . . . 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 3153	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ (Clsd‘𝐽)(𝐹 “ 𝑥) ∈ (Clsd‘𝐾))
31	30	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → ∀𝑥 ∈ (Clsd‘𝐽)(𝐹 “ 𝑥) ∈ (Clsd‘𝐾))
32		fofun 6773	. . . . . . . . . . 11 ⊢ (𝐹:∪ 𝐽–onto→𝑌 → Fun 𝐹)
33		funcnvcnv 6582	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → Fun ◡◡𝐹)
34		imadif 6599	. . . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → 𝐹:∪ 𝐽⟶𝑌)
37		fimacnv 6708	. . . . . . . . . . . 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 4079	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → ((◡𝐹 “ 𝑌) ∖ (◡𝐹 “ 𝑦)) = (∪ 𝐽 ∖ (◡𝐹 “ 𝑦)))
40	35, 39	eqtrd 2796	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (◡𝐹 “ (𝑌 ∖ 𝑦)) = (∪ 𝐽 ∖ (◡𝐹 “ 𝑦)))
41	7	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → 𝐽 ∈ Top)
42		simprr 782	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (◡𝐹 “ 𝑦) ∈ 𝐽)
43	15	opncld 23071	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ (◡𝐹 “ 𝑦) ∈ 𝐽) → (∪ 𝐽 ∖ (◡𝐹 “ 𝑦)) ∈ (Clsd‘𝐽))
44	41, 42, 43	syl2anc 593	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (∪ 𝐽 ∖ (◡𝐹 “ 𝑦)) ∈ (Clsd‘𝐽))
45	40, 44	eqeltrd 2861	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (◡𝐹 “ (𝑌 ∖ 𝑦)) ∈ (Clsd‘𝐽))
46	28, 31, 45	rspcdva 3582	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝐹 “ (◡𝐹 “ (𝑌 ∖ 𝑦))) ∈ (Clsd‘𝐾))
47	26, 46	eqeltrrd 2862	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (∪ 𝐾 ∖ 𝑦) ∈ (Clsd‘𝐾))
48		topontop 22951	. . . . . . . 8 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
49	22, 48	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → 𝐾 ∈ Top)
50		simprl 780	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → 𝑦 ⊆ 𝑌)
51	50, 24	sseqtrd 3972	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → 𝑦 ⊆ ∪ 𝐾)
52		eqid 2761	. . . . . . . 8 ⊢ ∪ 𝐾 = ∪ 𝐾
53	52	isopn2 23070	. . . . . . 7 ⊢ ((𝐾 ∈ Top ∧ 𝑦 ⊆ ∪ 𝐾) → (𝑦 ∈ 𝐾 ↔ (∪ 𝐾 ∖ 𝑦) ∈ (Clsd‘𝐾)))
54	49, 51, 53	syl2anc 593	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝑦 ∈ 𝐾 ↔ (∪ 𝐾 ∖ 𝑦) ∈ (Clsd‘𝐾)))
55	47, 54	mpbird 259	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → 𝑦 ∈ 𝐾)
56	55	ex 416	. . . 4 ⊢ (𝜑 → ((𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽) → 𝑦 ∈ 𝐾))
57	17, 56	sylbid 242	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐽 qTop 𝐹) → 𝑦 ∈ 𝐾))
58	57	ssrdv 3942	. 2 ⊢ (𝜑 → (𝐽 qTop 𝐹) ⊆ 𝐾)
59	5, 58	eqssd 3953	1 ⊢ (𝜑 → 𝐾 = (𝐽 qTop 𝐹))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075   ∖ cdif 3901   ⊆ wss 3904  ∪ cuni 4864  ^◡ccnv 5644  ran crn 5646   “ cima 5648  Fun wfun 6509   Fn wfn 6510  ⟶wf 6511  –onto→wfo 6513  ‘cfv 6515  (class class class)co 7390   qTop cqtop 17514  Topctop 22931  TopOnctopon 22948  Clsdccld 23054   Cn ccn 23262

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7712

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-ov 7393  df-oprab 7394  df-mpo 7395  df-map 8803  df-qtop 17518  df-top 22932  df-topon 22949  df-cld 23057  df-cn 23265

This theorem is referenced by: (None)