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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  ontowfo 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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