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Theorem qtopomap 22418
 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 22415 . . 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 21940 . . . . . . 7 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
71, 6syl 17 . . . . . 6 (𝜑𝐽 ∈ Top)
8 toptopon2 21618 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
97, 8sylib 221 . . . . 5 (𝜑𝐽 ∈ (TopOn‘ 𝐽))
10 cnf2 21949 . . . . . . . 8 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹: 𝐽𝑌)
119, 2, 1, 10syl3anc 1368 . . . . . . 7 (𝜑𝐹: 𝐽𝑌)
1211ffnd 6499 . . . . . 6 (𝜑𝐹 Fn 𝐽)
13 df-fo 6341 . . . . . 6 (𝐹: 𝐽onto𝑌 ↔ (𝐹 Fn 𝐽 ∧ ran 𝐹 = 𝑌))
1412, 3, 13sylanbrc 586 . . . . 5 (𝜑𝐹: 𝐽onto𝑌)
15 elqtop3 22403 . . . . 5 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐹: 𝐽onto𝑌) → (𝑦 ∈ (𝐽 qTop 𝐹) ↔ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)))
169, 14, 15syl2anc 587 . . . 4 (𝜑 → (𝑦 ∈ (𝐽 qTop 𝐹) ↔ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)))
17 foimacnv 6619 . . . . . . . 8 ((𝐹: 𝐽onto𝑌𝑦𝑌) → (𝐹 “ (𝐹𝑦)) = 𝑦)
1814, 17sylan 583 . . . . . . 7 ((𝜑𝑦𝑌) → (𝐹 “ (𝐹𝑦)) = 𝑦)
1918adantrr 716 . . . . . 6 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝐹𝑦)) = 𝑦)
20 imaeq2 5897 . . . . . . . 8 (𝑥 = (𝐹𝑦) → (𝐹𝑥) = (𝐹 “ (𝐹𝑦)))
2120eleq1d 2836 . . . . . . 7 (𝑥 = (𝐹𝑦) → ((𝐹𝑥) ∈ 𝐾 ↔ (𝐹 “ (𝐹𝑦)) ∈ 𝐾))
22 qtopomap.7 . . . . . . . . 9 ((𝜑𝑥𝐽) → (𝐹𝑥) ∈ 𝐾)
2322ralrimiva 3113 . . . . . . . 8 (𝜑 → ∀𝑥𝐽 (𝐹𝑥) ∈ 𝐾)
2423adantr 484 . . . . . . 7 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → ∀𝑥𝐽 (𝐹𝑥) ∈ 𝐾)
25 simprr 772 . . . . . . 7 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹𝑦) ∈ 𝐽)
2621, 24, 25rspcdva 3543 . . . . . 6 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝐹𝑦)) ∈ 𝐾)
2719, 26eqeltrrd 2853 . . . . 5 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝑦𝐾)
2827ex 416 . . . 4 (𝜑 → ((𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽) → 𝑦𝐾))
2916, 28sylbid 243 . . 3 (𝜑 → (𝑦 ∈ (𝐽 qTop 𝐹) → 𝑦𝐾))
3029ssrdv 3898 . 2 (𝜑 → (𝐽 qTop 𝐹) ⊆ 𝐾)
315, 30eqssd 3909 1 (𝜑𝐾 = (𝐽 qTop 𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070   ⊆ wss 3858  ∪ cuni 4798  ◡ccnv 5523  ran crn 5525   “ cima 5527   Fn wfn 6330  ⟶wf 6331  –onto→wfo 6333  ‘cfv 6335  (class class class)co 7150   qTop cqtop 16834  Topctop 21593  TopOnctopon 21610   Cn ccn 21924 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-map 8418  df-qtop 16838  df-top 21594  df-topon 21611  df-cn 21927 This theorem is referenced by:  hmeoqtop  22475
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