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Theorem qtopcmap 23714
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 23710 . . 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 23235 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
71, 6syl 17 . . . . 5 (𝜑𝐽 ∈ Top)
8 toptopon2 22911 . . . . . . . . 9 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
97, 8sylib 217 . . . . . . . 8 (𝜑𝐽 ∈ (TopOn‘ 𝐽))
10 cnf2 23244 . . . . . . . 8 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹: 𝐽𝑌)
119, 2, 1, 10syl3anc 1368 . . . . . . 7 (𝜑𝐹: 𝐽𝑌)
1211ffnd 6729 . . . . . 6 (𝜑𝐹 Fn 𝐽)
13 df-fo 6560 . . . . . 6 (𝐹: 𝐽onto𝑌 ↔ (𝐹 Fn 𝐽 ∧ ran 𝐹 = 𝑌))
1412, 3, 13sylanbrc 581 . . . . 5 (𝜑𝐹: 𝐽onto𝑌)
15 eqid 2726 . . . . . 6 𝐽 = 𝐽
1615elqtop2 23696 . . . . 5 ((𝐽 ∈ Top ∧ 𝐹: 𝐽onto𝑌) → (𝑦 ∈ (𝐽 qTop 𝐹) ↔ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)))
177, 14, 16syl2anc 582 . . . 4 (𝜑 → (𝑦 ∈ (𝐽 qTop 𝐹) ↔ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)))
1814adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝐹: 𝐽onto𝑌)
19 difss 4131 . . . . . . . . 9 (𝑌𝑦) ⊆ 𝑌
20 foimacnv 6860 . . . . . . . . 9 ((𝐹: 𝐽onto𝑌 ∧ (𝑌𝑦) ⊆ 𝑌) → (𝐹 “ (𝐹 “ (𝑌𝑦))) = (𝑌𝑦))
2118, 19, 20sylancl 584 . . . . . . . 8 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝐹 “ (𝑌𝑦))) = (𝑌𝑦))
222adantr 479 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝐾 ∈ (TopOn‘𝑌))
23 toponuni 22907 . . . . . . . . . 10 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
2422, 23syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝑌 = 𝐾)
2524difeq1d 4120 . . . . . . . 8 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝑌𝑦) = ( 𝐾𝑦))
2621, 25eqtrd 2766 . . . . . . 7 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝐹 “ (𝑌𝑦))) = ( 𝐾𝑦))
27 imaeq2 6065 . . . . . . . . 9 (𝑥 = (𝐹 “ (𝑌𝑦)) → (𝐹𝑥) = (𝐹 “ (𝐹 “ (𝑌𝑦))))
2827eleq1d 2811 . . . . . . . 8 (𝑥 = (𝐹 “ (𝑌𝑦)) → ((𝐹𝑥) ∈ (Clsd‘𝐾) ↔ (𝐹 “ (𝐹 “ (𝑌𝑦))) ∈ (Clsd‘𝐾)))
29 qtopcmap.7 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Clsd‘𝐽)) → (𝐹𝑥) ∈ (Clsd‘𝐾))
3029ralrimiva 3136 . . . . . . . . 9 (𝜑 → ∀𝑥 ∈ (Clsd‘𝐽)(𝐹𝑥) ∈ (Clsd‘𝐾))
3130adantr 479 . . . . . . . 8 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → ∀𝑥 ∈ (Clsd‘𝐽)(𝐹𝑥) ∈ (Clsd‘𝐾))
32 fofun 6816 . . . . . . . . . . 11 (𝐹: 𝐽onto𝑌 → Fun 𝐹)
33 funcnvcnv 6626 . . . . . . . . . . 11 (Fun 𝐹 → Fun 𝐹)
34 imadif 6643 . . . . . . . . . . 11 (Fun 𝐹 → (𝐹 “ (𝑌𝑦)) = ((𝐹𝑌) ∖ (𝐹𝑦)))
3518, 32, 33, 344syl 19 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝑌𝑦)) = ((𝐹𝑌) ∖ (𝐹𝑦)))
3611adantr 479 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝐹: 𝐽𝑌)
37 fimacnv 6750 . . . . . . . . . . . 12 (𝐹: 𝐽𝑌 → (𝐹𝑌) = 𝐽)
3836, 37syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹𝑌) = 𝐽)
3938difeq1d 4120 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → ((𝐹𝑌) ∖ (𝐹𝑦)) = ( 𝐽 ∖ (𝐹𝑦)))
4035, 39eqtrd 2766 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝑌𝑦)) = ( 𝐽 ∖ (𝐹𝑦)))
417adantr 479 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝐽 ∈ Top)
42 simprr 771 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹𝑦) ∈ 𝐽)
4315opncld 23028 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝐹𝑦) ∈ 𝐽) → ( 𝐽 ∖ (𝐹𝑦)) ∈ (Clsd‘𝐽))
4441, 42, 43syl2anc 582 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → ( 𝐽 ∖ (𝐹𝑦)) ∈ (Clsd‘𝐽))
4540, 44eqeltrd 2826 . . . . . . . 8 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝑌𝑦)) ∈ (Clsd‘𝐽))
4628, 31, 45rspcdva 3609 . . . . . . 7 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝐹 “ (𝑌𝑦))) ∈ (Clsd‘𝐾))
4726, 46eqeltrrd 2827 . . . . . 6 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → ( 𝐾𝑦) ∈ (Clsd‘𝐾))
48 topontop 22906 . . . . . . . 8 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
4922, 48syl 17 . . . . . . 7 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝐾 ∈ Top)
50 simprl 769 . . . . . . . 8 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝑦𝑌)
5150, 24sseqtrd 4020 . . . . . . 7 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝑦 𝐾)
52 eqid 2726 . . . . . . . 8 𝐾 = 𝐾
5352isopn2 23027 . . . . . . 7 ((𝐾 ∈ Top ∧ 𝑦 𝐾) → (𝑦𝐾 ↔ ( 𝐾𝑦) ∈ (Clsd‘𝐾)))
5449, 51, 53syl2anc 582 . . . . . 6 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝑦𝐾 ↔ ( 𝐾𝑦) ∈ (Clsd‘𝐾)))
5547, 54mpbird 256 . . . . 5 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝑦𝐾)
5655ex 411 . . . 4 (𝜑 → ((𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽) → 𝑦𝐾))
5717, 56sylbid 239 . . 3 (𝜑 → (𝑦 ∈ (𝐽 qTop 𝐹) → 𝑦𝐾))
5857ssrdv 3985 . 2 (𝜑 → (𝐽 qTop 𝐹) ⊆ 𝐾)
595, 58eqssd 3997 1 (𝜑𝐾 = (𝐽 qTop 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  wral 3051  cdif 3944  wss 3947   cuni 4913  ccnv 5681  ran crn 5683  cima 5685  Fun wfun 6548   Fn wfn 6549  wf 6550  ontowfo 6552  cfv 6554  (class class class)co 7424   qTop cqtop 17518  Topctop 22886  TopOnctopon 22903  Clsdccld 23011   Cn ccn 23219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-map 8857  df-qtop 17522  df-top 22887  df-topon 22904  df-cld 23014  df-cn 23222
This theorem is referenced by: (None)
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