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Theorem qtopt1 31499
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 22227 . . . 4 (𝐽 ∈ Fre → 𝐽 ∈ Top)
31, 2syl 17 . . 3 (𝜑𝐽 ∈ Top)
4 qtopt1.2 . . . 4 (𝜑𝐹:𝑋onto𝑌)
5 fofn 6635 . . . 4 (𝐹:𝑋onto𝑌𝐹 Fn 𝑋)
64, 5syl 17 . . 3 (𝜑𝐹 Fn 𝑋)
7 qtopt1.x . . . 4 𝑋 = 𝐽
87qtoptop 22597 . . 3 ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Top)
93, 6, 8syl2anc 587 . 2 (𝜑 → (𝐽 qTop 𝐹) ∈ Top)
10 simpr 488 . . . . . 6 ((𝜑𝑥 (𝐽 qTop 𝐹)) → 𝑥 (𝐽 qTop 𝐹))
117qtopuni 22599 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐹:𝑋onto𝑌) → 𝑌 = (𝐽 qTop 𝐹))
123, 4, 11syl2anc 587 . . . . . . 7 (𝜑𝑌 = (𝐽 qTop 𝐹))
1312adantr 484 . . . . . 6 ((𝜑𝑥 (𝐽 qTop 𝐹)) → 𝑌 = (𝐽 qTop 𝐹))
1410, 13eleqtrrd 2841 . . . . 5 ((𝜑𝑥 (𝐽 qTop 𝐹)) → 𝑥𝑌)
1514snssd 4722 . . . 4 ((𝜑𝑥 (𝐽 qTop 𝐹)) → {𝑥} ⊆ 𝑌)
16 qtopt1.3 . . . . 5 ((𝜑𝑥𝑌) → (𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))
1714, 16syldan 594 . . . 4 ((𝜑𝑥 (𝐽 qTop 𝐹)) → (𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))
183, 7jctir 524 . . . . . . 7 (𝜑 → (𝐽 ∈ Top ∧ 𝑋 = 𝐽))
19 istopon 21809 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝐽))
2018, 19sylibr 237 . . . . . 6 (𝜑𝐽 ∈ (TopOn‘𝑋))
21 qtopcld 22610 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → ({𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ({𝑥} ⊆ 𝑌 ∧ (𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))))
2220, 4, 21syl2anc 587 . . . . 5 (𝜑 → ({𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ({𝑥} ⊆ 𝑌 ∧ (𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))))
2322adantr 484 . . . 4 ((𝜑𝑥 (𝐽 qTop 𝐹)) → ({𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ({𝑥} ⊆ 𝑌 ∧ (𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))))
2415, 17, 23mpbir2and 713 . . 3 ((𝜑𝑥 (𝐽 qTop 𝐹)) → {𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)))
2524ralrimiva 3105 . 2 (𝜑 → ∀𝑥 (𝐽 qTop 𝐹){𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)))
26 eqid 2737 . . 3 (𝐽 qTop 𝐹) = (𝐽 qTop 𝐹)
2726ist1 22218 . 2 ((𝐽 qTop 𝐹) ∈ Fre ↔ ((𝐽 qTop 𝐹) ∈ Top ∧ ∀𝑥 (𝐽 qTop 𝐹){𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹))))
289, 25, 27sylanbrc 586 1 (𝜑 → (𝐽 qTop 𝐹) ∈ Fre)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  wral 3061  wss 3866  {csn 4541   cuni 4819  ccnv 5550  cima 5554   Fn wfn 6375  ontowfo 6378  cfv 6380  (class class class)co 7213   qTop cqtop 17008  Topctop 21790  TopOnctopon 21807  Clsdccld 21913  Frect1 22204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-qtop 17012  df-top 21791  df-topon 21808  df-cld 21916  df-t1 22211
This theorem is referenced by: (None)
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