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Theorem qtopcmap 21332
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 (𝜑𝐾 ∈ (TopOn‘𝑌))
qtopomap.5 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
qtopomap.6 (𝜑 → ran 𝐹 = 𝑌)
qtopcmap.7 ((𝜑𝑥 ∈ (Clsd‘𝐽)) → (𝐹𝑥) ∈ (Clsd‘𝐾))
Assertion
Ref Expression
qtopcmap (𝜑𝐾 = (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.
StepHypRef Expression
1 qtopomap.5 . . 3 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
2 qtopomap.4 . . 3 (𝜑𝐾 ∈ (TopOn‘𝑌))
3 qtopomap.6 . . 3 (𝜑 → ran 𝐹 = 𝑌)
4 qtopss 21328 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) → 𝐾 ⊆ (𝐽 qTop 𝐹))
51, 2, 3, 4syl3anc 1318 . 2 (𝜑𝐾 ⊆ (𝐽 qTop 𝐹))
6 cntop1 20854 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
71, 6syl 17 . . . . 5 (𝜑𝐽 ∈ Top)
8 eqid 2610 . . . . . . . . . 10 𝐽 = 𝐽
98toptopon 20548 . . . . . . . . 9 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
107, 9sylib 207 . . . . . . . 8 (𝜑𝐽 ∈ (TopOn‘ 𝐽))
11 cnf2 20863 . . . . . . . 8 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹: 𝐽𝑌)
1210, 2, 1, 11syl3anc 1318 . . . . . . 7 (𝜑𝐹: 𝐽𝑌)
13 ffn 5958 . . . . . . 7 (𝐹: 𝐽𝑌𝐹 Fn 𝐽)
1412, 13syl 17 . . . . . 6 (𝜑𝐹 Fn 𝐽)
15 df-fo 5810 . . . . . 6 (𝐹: 𝐽onto𝑌 ↔ (𝐹 Fn 𝐽 ∧ ran 𝐹 = 𝑌))
1614, 3, 15sylanbrc 695 . . . . 5 (𝜑𝐹: 𝐽onto𝑌)
178elqtop2 21314 . . . . 5 ((𝐽 ∈ Top ∧ 𝐹: 𝐽onto𝑌) → (𝑦 ∈ (𝐽 qTop 𝐹) ↔ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)))
187, 16, 17syl2anc 691 . . . 4 (𝜑 → (𝑦 ∈ (𝐽 qTop 𝐹) ↔ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)))
1916adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝐹: 𝐽onto𝑌)
20 difss 3699 . . . . . . . . 9 (𝑌𝑦) ⊆ 𝑌
21 foimacnv 6067 . . . . . . . . 9 ((𝐹: 𝐽onto𝑌 ∧ (𝑌𝑦) ⊆ 𝑌) → (𝐹 “ (𝐹 “ (𝑌𝑦))) = (𝑌𝑦))
2219, 20, 21sylancl 693 . . . . . . . 8 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝐹 “ (𝑌𝑦))) = (𝑌𝑦))
232adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝐾 ∈ (TopOn‘𝑌))
24 toponuni 20542 . . . . . . . . . 10 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
2523, 24syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝑌 = 𝐾)
2625difeq1d 3689 . . . . . . . 8 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝑌𝑦) = ( 𝐾𝑦))
2722, 26eqtrd 2644 . . . . . . 7 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝐹 “ (𝑌𝑦))) = ( 𝐾𝑦))
28 fofun 6029 . . . . . . . . . . 11 (𝐹: 𝐽onto𝑌 → Fun 𝐹)
29 funcnvcnv 5870 . . . . . . . . . . 11 (Fun 𝐹 → Fun 𝐹)
30 imadif 5887 . . . . . . . . . . 11 (Fun 𝐹 → (𝐹 “ (𝑌𝑦)) = ((𝐹𝑌) ∖ (𝐹𝑦)))
3119, 28, 29, 304syl 19 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝑌𝑦)) = ((𝐹𝑌) ∖ (𝐹𝑦)))
3212adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝐹: 𝐽𝑌)
33 fimacnv 6255 . . . . . . . . . . . 12 (𝐹: 𝐽𝑌 → (𝐹𝑌) = 𝐽)
3432, 33syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹𝑌) = 𝐽)
3534difeq1d 3689 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → ((𝐹𝑌) ∖ (𝐹𝑦)) = ( 𝐽 ∖ (𝐹𝑦)))
3631, 35eqtrd 2644 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝑌𝑦)) = ( 𝐽 ∖ (𝐹𝑦)))
377adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝐽 ∈ Top)
38 simprr 792 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹𝑦) ∈ 𝐽)
398opncld 20647 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝐹𝑦) ∈ 𝐽) → ( 𝐽 ∖ (𝐹𝑦)) ∈ (Clsd‘𝐽))
4037, 38, 39syl2anc 691 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → ( 𝐽 ∖ (𝐹𝑦)) ∈ (Clsd‘𝐽))
4136, 40eqeltrd 2688 . . . . . . . 8 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝑌𝑦)) ∈ (Clsd‘𝐽))
42 qtopcmap.7 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Clsd‘𝐽)) → (𝐹𝑥) ∈ (Clsd‘𝐾))
4342ralrimiva 2949 . . . . . . . . 9 (𝜑 → ∀𝑥 ∈ (Clsd‘𝐽)(𝐹𝑥) ∈ (Clsd‘𝐾))
4443adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → ∀𝑥 ∈ (Clsd‘𝐽)(𝐹𝑥) ∈ (Clsd‘𝐾))
45 imaeq2 5381 . . . . . . . . . 10 (𝑥 = (𝐹 “ (𝑌𝑦)) → (𝐹𝑥) = (𝐹 “ (𝐹 “ (𝑌𝑦))))
4645eleq1d 2672 . . . . . . . . 9 (𝑥 = (𝐹 “ (𝑌𝑦)) → ((𝐹𝑥) ∈ (Clsd‘𝐾) ↔ (𝐹 “ (𝐹 “ (𝑌𝑦))) ∈ (Clsd‘𝐾)))
4746rspcv 3278 . . . . . . . 8 ((𝐹 “ (𝑌𝑦)) ∈ (Clsd‘𝐽) → (∀𝑥 ∈ (Clsd‘𝐽)(𝐹𝑥) ∈ (Clsd‘𝐾) → (𝐹 “ (𝐹 “ (𝑌𝑦))) ∈ (Clsd‘𝐾)))
4841, 44, 47sylc 63 . . . . . . 7 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝐹 “ (𝑌𝑦))) ∈ (Clsd‘𝐾))
4927, 48eqeltrrd 2689 . . . . . 6 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → ( 𝐾𝑦) ∈ (Clsd‘𝐾))
50 topontop 20541 . . . . . . . 8 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
5123, 50syl 17 . . . . . . 7 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝐾 ∈ Top)
52 simprl 790 . . . . . . . 8 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝑦𝑌)
5352, 25sseqtrd 3604 . . . . . . 7 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝑦 𝐾)
54 eqid 2610 . . . . . . . 8 𝐾 = 𝐾
5554isopn2 20646 . . . . . . 7 ((𝐾 ∈ Top ∧ 𝑦 𝐾) → (𝑦𝐾 ↔ ( 𝐾𝑦) ∈ (Clsd‘𝐾)))
5651, 53, 55syl2anc 691 . . . . . 6 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝑦𝐾 ↔ ( 𝐾𝑦) ∈ (Clsd‘𝐾)))
5749, 56mpbird 246 . . . . 5 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝑦𝐾)
5857ex 449 . . . 4 (𝜑 → ((𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽) → 𝑦𝐾))
5918, 58sylbid 229 . . 3 (𝜑 → (𝑦 ∈ (𝐽 qTop 𝐹) → 𝑦𝐾))
6059ssrdv 3574 . 2 (𝜑 → (𝐽 qTop 𝐹) ⊆ 𝐾)
615, 60eqssd 3585 1 (𝜑𝐾 = (𝐽 qTop 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  cdif 3537  wss 3540   cuni 4372  ccnv 5037  ran crn 5039  cima 5041  Fun wfun 5798   Fn wfn 5799  wf 5800  ontowfo 5802  cfv 5804  (class class class)co 6549   qTop cqtop 15986  Topctop 20517  TopOnctopon 20518  Clsdccld 20630   Cn ccn 20838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-qtop 15990  df-top 20521  df-topon 20523  df-cld 20633  df-cn 20841
This theorem is referenced by: (None)
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