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Theorem qtopomap 23727
Description: If 𝐹 is a surjective continuous open map, then it is a quotient map. (An open map is a function that maps open sets to open 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)
qtopomap.7 ((𝜑𝑥𝐽) → (𝐹𝑥) ∈ 𝐾)
Assertion
Ref Expression
qtopomap (𝜑𝐾 = (𝐽 qTop 𝐹))
Distinct variable groups:   𝑥,𝐹   𝑥,𝐽   𝑥,𝐾   𝜑,𝑥   𝑥,𝑌

Proof of Theorem qtopomap
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 23724 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) → 𝐾 ⊆ (𝐽 qTop 𝐹))
51, 2, 3, 4syl3anc 1372 . 2 (𝜑𝐾 ⊆ (𝐽 qTop 𝐹))
6 cntop1 23249 . . . . . . 7 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
71, 6syl 17 . . . . . 6 (𝜑𝐽 ∈ Top)
8 toptopon2 22925 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
97, 8sylib 218 . . . . 5 (𝜑𝐽 ∈ (TopOn‘ 𝐽))
10 cnf2 23258 . . . . . . . 8 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹: 𝐽𝑌)
119, 2, 1, 10syl3anc 1372 . . . . . . 7 (𝜑𝐹: 𝐽𝑌)
1211ffnd 6736 . . . . . 6 (𝜑𝐹 Fn 𝐽)
13 df-fo 6566 . . . . . 6 (𝐹: 𝐽onto𝑌 ↔ (𝐹 Fn 𝐽 ∧ ran 𝐹 = 𝑌))
1412, 3, 13sylanbrc 583 . . . . 5 (𝜑𝐹: 𝐽onto𝑌)
15 elqtop3 23712 . . . . 5 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐹: 𝐽onto𝑌) → (𝑦 ∈ (𝐽 qTop 𝐹) ↔ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)))
169, 14, 15syl2anc 584 . . . 4 (𝜑 → (𝑦 ∈ (𝐽 qTop 𝐹) ↔ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)))
17 foimacnv 6864 . . . . . . . 8 ((𝐹: 𝐽onto𝑌𝑦𝑌) → (𝐹 “ (𝐹𝑦)) = 𝑦)
1814, 17sylan 580 . . . . . . 7 ((𝜑𝑦𝑌) → (𝐹 “ (𝐹𝑦)) = 𝑦)
1918adantrr 717 . . . . . 6 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝐹𝑦)) = 𝑦)
20 imaeq2 6073 . . . . . . . 8 (𝑥 = (𝐹𝑦) → (𝐹𝑥) = (𝐹 “ (𝐹𝑦)))
2120eleq1d 2825 . . . . . . 7 (𝑥 = (𝐹𝑦) → ((𝐹𝑥) ∈ 𝐾 ↔ (𝐹 “ (𝐹𝑦)) ∈ 𝐾))
22 qtopomap.7 . . . . . . . . 9 ((𝜑𝑥𝐽) → (𝐹𝑥) ∈ 𝐾)
2322ralrimiva 3145 . . . . . . . 8 (𝜑 → ∀𝑥𝐽 (𝐹𝑥) ∈ 𝐾)
2423adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → ∀𝑥𝐽 (𝐹𝑥) ∈ 𝐾)
25 simprr 772 . . . . . . 7 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹𝑦) ∈ 𝐽)
2621, 24, 25rspcdva 3622 . . . . . 6 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝐹𝑦)) ∈ 𝐾)
2719, 26eqeltrrd 2841 . . . . 5 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝑦𝐾)
2827ex 412 . . . 4 (𝜑 → ((𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽) → 𝑦𝐾))
2916, 28sylbid 240 . . 3 (𝜑 → (𝑦 ∈ (𝐽 qTop 𝐹) → 𝑦𝐾))
3029ssrdv 3988 . 2 (𝜑 → (𝐽 qTop 𝐹) ⊆ 𝐾)
315, 30eqssd 4000 1 (𝜑𝐾 = (𝐽 qTop 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  wral 3060  wss 3950   cuni 4906  ccnv 5683  ran crn 5685  cima 5687   Fn wfn 6555  wf 6556  ontowfo 6558  cfv 6560  (class class class)co 7432   qTop cqtop 17549  Topctop 22900  TopOnctopon 22917   Cn ccn 23233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-map 8869  df-qtop 17553  df-top 22901  df-topon 22918  df-cn 23236
This theorem is referenced by:  hmeoqtop  23784
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