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Theorem qtopcmap 22941
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 22937 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) → 𝐾 ⊆ (𝐽 qTop 𝐹))
51, 2, 3, 4syl3anc 1370 . 2 (𝜑𝐾 ⊆ (𝐽 qTop 𝐹))
6 cntop1 22462 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
71, 6syl 17 . . . . 5 (𝜑𝐽 ∈ Top)
8 toptopon2 22138 . . . . . . . . 9 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
97, 8sylib 217 . . . . . . . 8 (𝜑𝐽 ∈ (TopOn‘ 𝐽))
10 cnf2 22471 . . . . . . . 8 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹: 𝐽𝑌)
119, 2, 1, 10syl3anc 1370 . . . . . . 7 (𝜑𝐹: 𝐽𝑌)
1211ffnd 6636 . . . . . 6 (𝜑𝐹 Fn 𝐽)
13 df-fo 6469 . . . . . 6 (𝐹: 𝐽onto𝑌 ↔ (𝐹 Fn 𝐽 ∧ ran 𝐹 = 𝑌))
1412, 3, 13sylanbrc 583 . . . . 5 (𝜑𝐹: 𝐽onto𝑌)
15 eqid 2737 . . . . . 6 𝐽 = 𝐽
1615elqtop2 22923 . . . . 5 ((𝐽 ∈ Top ∧ 𝐹: 𝐽onto𝑌) → (𝑦 ∈ (𝐽 qTop 𝐹) ↔ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)))
177, 14, 16syl2anc 584 . . . 4 (𝜑 → (𝑦 ∈ (𝐽 qTop 𝐹) ↔ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)))
1814adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝐹: 𝐽onto𝑌)
19 difss 4076 . . . . . . . . 9 (𝑌𝑦) ⊆ 𝑌
20 foimacnv 6768 . . . . . . . . 9 ((𝐹: 𝐽onto𝑌 ∧ (𝑌𝑦) ⊆ 𝑌) → (𝐹 “ (𝐹 “ (𝑌𝑦))) = (𝑌𝑦))
2118, 19, 20sylancl 586 . . . . . . . 8 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝐹 “ (𝑌𝑦))) = (𝑌𝑦))
222adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝐾 ∈ (TopOn‘𝑌))
23 toponuni 22134 . . . . . . . . . 10 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
2422, 23syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝑌 = 𝐾)
2524difeq1d 4066 . . . . . . . 8 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝑌𝑦) = ( 𝐾𝑦))
2621, 25eqtrd 2777 . . . . . . 7 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝐹 “ (𝑌𝑦))) = ( 𝐾𝑦))
27 imaeq2 5980 . . . . . . . . 9 (𝑥 = (𝐹 “ (𝑌𝑦)) → (𝐹𝑥) = (𝐹 “ (𝐹 “ (𝑌𝑦))))
2827eleq1d 2822 . . . . . . . 8 (𝑥 = (𝐹 “ (𝑌𝑦)) → ((𝐹𝑥) ∈ (Clsd‘𝐾) ↔ (𝐹 “ (𝐹 “ (𝑌𝑦))) ∈ (Clsd‘𝐾)))
29 qtopcmap.7 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Clsd‘𝐽)) → (𝐹𝑥) ∈ (Clsd‘𝐾))
3029ralrimiva 3140 . . . . . . . . 9 (𝜑 → ∀𝑥 ∈ (Clsd‘𝐽)(𝐹𝑥) ∈ (Clsd‘𝐾))
3130adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → ∀𝑥 ∈ (Clsd‘𝐽)(𝐹𝑥) ∈ (Clsd‘𝐾))
32 fofun 6724 . . . . . . . . . . 11 (𝐹: 𝐽onto𝑌 → Fun 𝐹)
33 funcnvcnv 6535 . . . . . . . . . . 11 (Fun 𝐹 → Fun 𝐹)
34 imadif 6552 . . . . . . . . . . 11 (Fun 𝐹 → (𝐹 “ (𝑌𝑦)) = ((𝐹𝑌) ∖ (𝐹𝑦)))
3518, 32, 33, 344syl 19 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝑌𝑦)) = ((𝐹𝑌) ∖ (𝐹𝑦)))
3611adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝐹: 𝐽𝑌)
37 fimacnv 6657 . . . . . . . . . . . 12 (𝐹: 𝐽𝑌 → (𝐹𝑌) = 𝐽)
3836, 37syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹𝑌) = 𝐽)
3938difeq1d 4066 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → ((𝐹𝑌) ∖ (𝐹𝑦)) = ( 𝐽 ∖ (𝐹𝑦)))
4035, 39eqtrd 2777 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝑌𝑦)) = ( 𝐽 ∖ (𝐹𝑦)))
417adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝐽 ∈ Top)
42 simprr 770 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹𝑦) ∈ 𝐽)
4315opncld 22255 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝐹𝑦) ∈ 𝐽) → ( 𝐽 ∖ (𝐹𝑦)) ∈ (Clsd‘𝐽))
4441, 42, 43syl2anc 584 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → ( 𝐽 ∖ (𝐹𝑦)) ∈ (Clsd‘𝐽))
4540, 44eqeltrd 2838 . . . . . . . 8 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝑌𝑦)) ∈ (Clsd‘𝐽))
4628, 31, 45rspcdva 3571 . . . . . . 7 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝐹 “ (𝑌𝑦))) ∈ (Clsd‘𝐾))
4726, 46eqeltrrd 2839 . . . . . 6 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → ( 𝐾𝑦) ∈ (Clsd‘𝐾))
48 topontop 22133 . . . . . . . 8 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
4922, 48syl 17 . . . . . . 7 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝐾 ∈ Top)
50 simprl 768 . . . . . . . 8 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝑦𝑌)
5150, 24sseqtrd 3970 . . . . . . 7 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝑦 𝐾)
52 eqid 2737 . . . . . . . 8 𝐾 = 𝐾
5352isopn2 22254 . . . . . . 7 ((𝐾 ∈ Top ∧ 𝑦 𝐾) → (𝑦𝐾 ↔ ( 𝐾𝑦) ∈ (Clsd‘𝐾)))
5449, 51, 53syl2anc 584 . . . . . 6 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝑦𝐾 ↔ ( 𝐾𝑦) ∈ (Clsd‘𝐾)))
5547, 54mpbird 256 . . . . 5 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝑦𝐾)
5655ex 413 . . . 4 (𝜑 → ((𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽) → 𝑦𝐾))
5717, 56sylbid 239 . . 3 (𝜑 → (𝑦 ∈ (𝐽 qTop 𝐹) → 𝑦𝐾))
5857ssrdv 3936 . 2 (𝜑 → (𝐽 qTop 𝐹) ⊆ 𝐾)
595, 58eqssd 3947 1 (𝜑𝐾 = (𝐽 qTop 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1540  wcel 2105  wral 3062  cdif 3893  wss 3896   cuni 4848  ccnv 5604  ran crn 5606  cima 5608  Fun wfun 6457   Fn wfn 6458  wf 6459  ontowfo 6461  cfv 6463  (class class class)co 7313   qTop cqtop 17281  Topctop 22113  TopOnctopon 22130  Clsdccld 22238   Cn ccn 22446
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5222  ax-sep 5236  ax-nul 5243  ax-pow 5301  ax-pr 5365  ax-un 7626
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4470  df-pw 4545  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4849  df-iun 4937  df-br 5086  df-opab 5148  df-mpt 5169  df-id 5505  df-xp 5611  df-rel 5612  df-cnv 5613  df-co 5614  df-dm 5615  df-rn 5616  df-res 5617  df-ima 5618  df-iota 6415  df-fun 6465  df-fn 6466  df-f 6467  df-f1 6468  df-fo 6469  df-f1o 6470  df-fv 6471  df-ov 7316  df-oprab 7317  df-mpo 7318  df-map 8663  df-qtop 17285  df-top 22114  df-topon 22131  df-cld 22241  df-cn 22449
This theorem is referenced by: (None)
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