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Theorem qtopcmap 22313
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 22309 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) → 𝐾 ⊆ (𝐽 qTop 𝐹))
51, 2, 3, 4syl3anc 1368 . 2 (𝜑𝐾 ⊆ (𝐽 qTop 𝐹))
6 cntop1 21834 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
71, 6syl 17 . . . . 5 (𝜑𝐽 ∈ Top)
8 toptopon2 21512 . . . . . . . . 9 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
97, 8sylib 221 . . . . . . . 8 (𝜑𝐽 ∈ (TopOn‘ 𝐽))
10 cnf2 21843 . . . . . . . 8 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹: 𝐽𝑌)
119, 2, 1, 10syl3anc 1368 . . . . . . 7 (𝜑𝐹: 𝐽𝑌)
1211ffnd 6496 . . . . . 6 (𝜑𝐹 Fn 𝐽)
13 df-fo 6342 . . . . . 6 (𝐹: 𝐽onto𝑌 ↔ (𝐹 Fn 𝐽 ∧ ran 𝐹 = 𝑌))
1412, 3, 13sylanbrc 586 . . . . 5 (𝜑𝐹: 𝐽onto𝑌)
15 eqid 2824 . . . . . 6 𝐽 = 𝐽
1615elqtop2 22295 . . . . 5 ((𝐽 ∈ Top ∧ 𝐹: 𝐽onto𝑌) → (𝑦 ∈ (𝐽 qTop 𝐹) ↔ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)))
177, 14, 16syl2anc 587 . . . 4 (𝜑 → (𝑦 ∈ (𝐽 qTop 𝐹) ↔ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)))
1814adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝐹: 𝐽onto𝑌)
19 difss 4092 . . . . . . . . 9 (𝑌𝑦) ⊆ 𝑌
20 foimacnv 6613 . . . . . . . . 9 ((𝐹: 𝐽onto𝑌 ∧ (𝑌𝑦) ⊆ 𝑌) → (𝐹 “ (𝐹 “ (𝑌𝑦))) = (𝑌𝑦))
2118, 19, 20sylancl 589 . . . . . . . 8 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝐹 “ (𝑌𝑦))) = (𝑌𝑦))
222adantr 484 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝐾 ∈ (TopOn‘𝑌))
23 toponuni 21508 . . . . . . . . . 10 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
2422, 23syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝑌 = 𝐾)
2524difeq1d 4082 . . . . . . . 8 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝑌𝑦) = ( 𝐾𝑦))
2621, 25eqtrd 2859 . . . . . . 7 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝐹 “ (𝑌𝑦))) = ( 𝐾𝑦))
27 imaeq2 5906 . . . . . . . . 9 (𝑥 = (𝐹 “ (𝑌𝑦)) → (𝐹𝑥) = (𝐹 “ (𝐹 “ (𝑌𝑦))))
2827eleq1d 2900 . . . . . . . 8 (𝑥 = (𝐹 “ (𝑌𝑦)) → ((𝐹𝑥) ∈ (Clsd‘𝐾) ↔ (𝐹 “ (𝐹 “ (𝑌𝑦))) ∈ (Clsd‘𝐾)))
29 qtopcmap.7 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Clsd‘𝐽)) → (𝐹𝑥) ∈ (Clsd‘𝐾))
3029ralrimiva 3176 . . . . . . . . 9 (𝜑 → ∀𝑥 ∈ (Clsd‘𝐽)(𝐹𝑥) ∈ (Clsd‘𝐾))
3130adantr 484 . . . . . . . 8 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → ∀𝑥 ∈ (Clsd‘𝐽)(𝐹𝑥) ∈ (Clsd‘𝐾))
32 fofun 6572 . . . . . . . . . . 11 (𝐹: 𝐽onto𝑌 → Fun 𝐹)
33 funcnvcnv 6402 . . . . . . . . . . 11 (Fun 𝐹 → Fun 𝐹)
34 imadif 6419 . . . . . . . . . . 11 (Fun 𝐹 → (𝐹 “ (𝑌𝑦)) = ((𝐹𝑌) ∖ (𝐹𝑦)))
3518, 32, 33, 344syl 19 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝑌𝑦)) = ((𝐹𝑌) ∖ (𝐹𝑦)))
3611adantr 484 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝐹: 𝐽𝑌)
37 fimacnv 6820 . . . . . . . . . . . 12 (𝐹: 𝐽𝑌 → (𝐹𝑌) = 𝐽)
3836, 37syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹𝑌) = 𝐽)
3938difeq1d 4082 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → ((𝐹𝑌) ∖ (𝐹𝑦)) = ( 𝐽 ∖ (𝐹𝑦)))
4035, 39eqtrd 2859 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝑌𝑦)) = ( 𝐽 ∖ (𝐹𝑦)))
417adantr 484 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝐽 ∈ Top)
42 simprr 772 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹𝑦) ∈ 𝐽)
4315opncld 21627 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝐹𝑦) ∈ 𝐽) → ( 𝐽 ∖ (𝐹𝑦)) ∈ (Clsd‘𝐽))
4441, 42, 43syl2anc 587 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → ( 𝐽 ∖ (𝐹𝑦)) ∈ (Clsd‘𝐽))
4540, 44eqeltrd 2916 . . . . . . . 8 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝑌𝑦)) ∈ (Clsd‘𝐽))
4628, 31, 45rspcdva 3610 . . . . . . 7 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝐹 “ (𝑌𝑦))) ∈ (Clsd‘𝐾))
4726, 46eqeltrrd 2917 . . . . . 6 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → ( 𝐾𝑦) ∈ (Clsd‘𝐾))
48 topontop 21507 . . . . . . . 8 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
4922, 48syl 17 . . . . . . 7 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝐾 ∈ Top)
50 simprl 770 . . . . . . . 8 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝑦𝑌)
5150, 24sseqtrd 3991 . . . . . . 7 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝑦 𝐾)
52 eqid 2824 . . . . . . . 8 𝐾 = 𝐾
5352isopn2 21626 . . . . . . 7 ((𝐾 ∈ Top ∧ 𝑦 𝐾) → (𝑦𝐾 ↔ ( 𝐾𝑦) ∈ (Clsd‘𝐾)))
5449, 51, 53syl2anc 587 . . . . . 6 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝑦𝐾 ↔ ( 𝐾𝑦) ∈ (Clsd‘𝐾)))
5547, 54mpbird 260 . . . . 5 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝑦𝐾)
5655ex 416 . . . 4 (𝜑 → ((𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽) → 𝑦𝐾))
5717, 56sylbid 243 . . 3 (𝜑 → (𝑦 ∈ (𝐽 qTop 𝐹) → 𝑦𝐾))
5857ssrdv 3957 . 2 (𝜑 → (𝐽 qTop 𝐹) ⊆ 𝐾)
595, 58eqssd 3968 1 (𝜑𝐾 = (𝐽 qTop 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3132  cdif 3915  wss 3918   cuni 4819  ccnv 5535  ran crn 5537  cima 5539  Fun wfun 6330   Fn wfn 6331  wf 6332  ontowfo 6334  cfv 6336  (class class class)co 7138   qTop cqtop 16765  Topctop 21487  TopOnctopon 21504  Clsdccld 21610   Cn ccn 21818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7141  df-oprab 7142  df-mpo 7143  df-map 8391  df-qtop 16769  df-top 21488  df-topon 21505  df-cld 21613  df-cn 21821
This theorem is referenced by: (None)
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