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Theorem qtopt1 29230
 Description: If every equivalence class is closed, then the quotient space is T1 . (Contributed by Thierry Arnoux, 5-Jan-2020.)
Hypotheses
Ref Expression
qtopt1.x 𝑋 = 𝐽
qtopt1.1 (𝜑𝐽 ∈ Fre)
qtopt1.2 (𝜑𝐹:𝑋onto𝑌)
qtopt1.3 ((𝜑𝑥𝑌) → (𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))
Assertion
Ref Expression
qtopt1 (𝜑 → (𝐽 qTop 𝐹) ∈ Fre)
Distinct variable groups:   𝑥,𝐹   𝑥,𝐽   𝜑,𝑥
Allowed substitution hints:   𝑋(𝑥)   𝑌(𝑥)

Proof of Theorem qtopt1
StepHypRef Expression
1 qtopt1.1 . . . 4 (𝜑𝐽 ∈ Fre)
2 t1top 20944 . . . 4 (𝐽 ∈ Fre → 𝐽 ∈ Top)
31, 2syl 17 . . 3 (𝜑𝐽 ∈ Top)
4 qtopt1.2 . . . 4 (𝜑𝐹:𝑋onto𝑌)
5 fofn 6030 . . . 4 (𝐹:𝑋onto𝑌𝐹 Fn 𝑋)
64, 5syl 17 . . 3 (𝜑𝐹 Fn 𝑋)
7 qtopt1.x . . . 4 𝑋 = 𝐽
87qtoptop 21313 . . 3 ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Top)
93, 6, 8syl2anc 691 . 2 (𝜑 → (𝐽 qTop 𝐹) ∈ Top)
10 simpr 476 . . . . . 6 ((𝜑𝑥 (𝐽 qTop 𝐹)) → 𝑥 (𝐽 qTop 𝐹))
117qtopuni 21315 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐹:𝑋onto𝑌) → 𝑌 = (𝐽 qTop 𝐹))
123, 4, 11syl2anc 691 . . . . . . 7 (𝜑𝑌 = (𝐽 qTop 𝐹))
1312adantr 480 . . . . . 6 ((𝜑𝑥 (𝐽 qTop 𝐹)) → 𝑌 = (𝐽 qTop 𝐹))
1410, 13eleqtrrd 2691 . . . . 5 ((𝜑𝑥 (𝐽 qTop 𝐹)) → 𝑥𝑌)
1514snssd 4281 . . . 4 ((𝜑𝑥 (𝐽 qTop 𝐹)) → {𝑥} ⊆ 𝑌)
16 qtopt1.3 . . . . 5 ((𝜑𝑥𝑌) → (𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))
1714, 16syldan 486 . . . 4 ((𝜑𝑥 (𝐽 qTop 𝐹)) → (𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))
183, 7jctir 559 . . . . . . 7 (𝜑 → (𝐽 ∈ Top ∧ 𝑋 = 𝐽))
19 istopon 20540 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝐽))
2018, 19sylibr 223 . . . . . 6 (𝜑𝐽 ∈ (TopOn‘𝑋))
21 qtopcld 21326 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → ({𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ({𝑥} ⊆ 𝑌 ∧ (𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))))
2220, 4, 21syl2anc 691 . . . . 5 (𝜑 → ({𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ({𝑥} ⊆ 𝑌 ∧ (𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))))
2322adantr 480 . . . 4 ((𝜑𝑥 (𝐽 qTop 𝐹)) → ({𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ({𝑥} ⊆ 𝑌 ∧ (𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))))
2415, 17, 23mpbir2and 959 . . 3 ((𝜑𝑥 (𝐽 qTop 𝐹)) → {𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)))
2524ralrimiva 2949 . 2 (𝜑 → ∀𝑥 (𝐽 qTop 𝐹){𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)))
26 eqid 2610 . . 3 (𝐽 qTop 𝐹) = (𝐽 qTop 𝐹)
2726ist1 20935 . 2 ((𝐽 qTop 𝐹) ∈ Fre ↔ ((𝐽 qTop 𝐹) ∈ Top ∧ ∀𝑥 (𝐽 qTop 𝐹){𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹))))
289, 25, 27sylanbrc 695 1 (𝜑 → (𝐽 qTop 𝐹) ∈ Fre)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540  {csn 4125  ∪ cuni 4372  ◡ccnv 5037   “ cima 5041   Fn wfn 5799  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549   qTop cqtop 15986  Topctop 20517  TopOnctopon 20518  Clsdccld 20630  Frect1 20921 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-qtop 15990  df-top 20521  df-topon 20523  df-cld 20633  df-t1 20928 This theorem is referenced by: (None)
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