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Theorem qtopomap 21569
 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 21566 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) → 𝐾 ⊆ (𝐽 qTop 𝐹))
51, 2, 3, 4syl3anc 1366 . 2 (𝜑𝐾 ⊆ (𝐽 qTop 𝐹))
6 cntop1 21092 . . . . . . 7 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
71, 6syl 17 . . . . . 6 (𝜑𝐽 ∈ Top)
8 eqid 2651 . . . . . . 7 𝐽 = 𝐽
98toptopon 20770 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
107, 9sylib 208 . . . . 5 (𝜑𝐽 ∈ (TopOn‘ 𝐽))
11 cnf2 21101 . . . . . . . 8 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹: 𝐽𝑌)
1210, 2, 1, 11syl3anc 1366 . . . . . . 7 (𝜑𝐹: 𝐽𝑌)
13 ffn 6083 . . . . . . 7 (𝐹: 𝐽𝑌𝐹 Fn 𝐽)
1412, 13syl 17 . . . . . 6 (𝜑𝐹 Fn 𝐽)
15 df-fo 5932 . . . . . 6 (𝐹: 𝐽onto𝑌 ↔ (𝐹 Fn 𝐽 ∧ ran 𝐹 = 𝑌))
1614, 3, 15sylanbrc 699 . . . . 5 (𝜑𝐹: 𝐽onto𝑌)
17 elqtop3 21554 . . . . 5 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐹: 𝐽onto𝑌) → (𝑦 ∈ (𝐽 qTop 𝐹) ↔ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)))
1810, 16, 17syl2anc 694 . . . 4 (𝜑 → (𝑦 ∈ (𝐽 qTop 𝐹) ↔ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)))
19 foimacnv 6192 . . . . . . . 8 ((𝐹: 𝐽onto𝑌𝑦𝑌) → (𝐹 “ (𝐹𝑦)) = 𝑦)
2016, 19sylan 487 . . . . . . 7 ((𝜑𝑦𝑌) → (𝐹 “ (𝐹𝑦)) = 𝑦)
2120adantrr 753 . . . . . 6 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝐹𝑦)) = 𝑦)
22 simprr 811 . . . . . . 7 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹𝑦) ∈ 𝐽)
23 qtopomap.7 . . . . . . . . 9 ((𝜑𝑥𝐽) → (𝐹𝑥) ∈ 𝐾)
2423ralrimiva 2995 . . . . . . . 8 (𝜑 → ∀𝑥𝐽 (𝐹𝑥) ∈ 𝐾)
2524adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → ∀𝑥𝐽 (𝐹𝑥) ∈ 𝐾)
26 imaeq2 5497 . . . . . . . . 9 (𝑥 = (𝐹𝑦) → (𝐹𝑥) = (𝐹 “ (𝐹𝑦)))
2726eleq1d 2715 . . . . . . . 8 (𝑥 = (𝐹𝑦) → ((𝐹𝑥) ∈ 𝐾 ↔ (𝐹 “ (𝐹𝑦)) ∈ 𝐾))
2827rspcv 3336 . . . . . . 7 ((𝐹𝑦) ∈ 𝐽 → (∀𝑥𝐽 (𝐹𝑥) ∈ 𝐾 → (𝐹 “ (𝐹𝑦)) ∈ 𝐾))
2922, 25, 28sylc 65 . . . . . 6 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → (𝐹 “ (𝐹𝑦)) ∈ 𝐾)
3021, 29eqeltrrd 2731 . . . . 5 ((𝜑 ∧ (𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽)) → 𝑦𝐾)
3130ex 449 . . . 4 (𝜑 → ((𝑦𝑌 ∧ (𝐹𝑦) ∈ 𝐽) → 𝑦𝐾))
3218, 31sylbid 230 . . 3 (𝜑 → (𝑦 ∈ (𝐽 qTop 𝐹) → 𝑦𝐾))
3332ssrdv 3642 . 2 (𝜑 → (𝐽 qTop 𝐹) ⊆ 𝐾)
345, 33eqssd 3653 1 (𝜑𝐾 = (𝐽 qTop 𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941   ⊆ wss 3607  ∪ cuni 4468  ◡ccnv 5142  ran crn 5144   “ cima 5146   Fn wfn 5921  ⟶wf 5922  –onto→wfo 5924  ‘cfv 5926  (class class class)co 6690   qTop cqtop 16210  Topctop 20746  TopOnctopon 20763   Cn ccn 21076 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-qtop 16214  df-top 20747  df-topon 20764  df-cn 21079 This theorem is referenced by:  hmeoqtop  21626
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