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Theorem qtopt1 29026
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 20847 . . . 4 (𝐽 ∈ Fre → 𝐽 ∈ Top)
31, 2syl 17 . . 3 (𝜑𝐽 ∈ Top)
4 qtopt1.2 . . . 4 (𝜑𝐹:𝑋onto𝑌)
5 fofn 5914 . . . 4 (𝐹:𝑋onto𝑌𝐹 Fn 𝑋)
64, 5syl 17 . . 3 (𝜑𝐹 Fn 𝑋)
7 qtopt1.x . . . 4 𝑋 = 𝐽
87qtoptop 21216 . . 3 ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Top)
93, 6, 8syl2anc 690 . 2 (𝜑 → (𝐽 qTop 𝐹) ∈ Top)
10 simpr 475 . . . . . 6 ((𝜑𝑥 (𝐽 qTop 𝐹)) → 𝑥 (𝐽 qTop 𝐹))
117qtopuni 21218 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐹:𝑋onto𝑌) → 𝑌 = (𝐽 qTop 𝐹))
123, 4, 11syl2anc 690 . . . . . . 7 (𝜑𝑌 = (𝐽 qTop 𝐹))
1312adantr 479 . . . . . 6 ((𝜑𝑥 (𝐽 qTop 𝐹)) → 𝑌 = (𝐽 qTop 𝐹))
1410, 13eleqtrrd 2595 . . . . 5 ((𝜑𝑥 (𝐽 qTop 𝐹)) → 𝑥𝑌)
1514snssd 4184 . . . 4 ((𝜑𝑥 (𝐽 qTop 𝐹)) → {𝑥} ⊆ 𝑌)
16 qtopt1.3 . . . . 5 ((𝜑𝑥𝑌) → (𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))
1714, 16syldan 485 . . . 4 ((𝜑𝑥 (𝐽 qTop 𝐹)) → (𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))
183, 7jctir 558 . . . . . . 7 (𝜑 → (𝐽 ∈ Top ∧ 𝑋 = 𝐽))
19 istopon 20443 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝐽))
2018, 19sylibr 222 . . . . . 6 (𝜑𝐽 ∈ (TopOn‘𝑋))
21 qtopcld 21229 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → ({𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ({𝑥} ⊆ 𝑌 ∧ (𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))))
2220, 4, 21syl2anc 690 . . . . 5 (𝜑 → ({𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ({𝑥} ⊆ 𝑌 ∧ (𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))))
2322adantr 479 . . . 4 ((𝜑𝑥 (𝐽 qTop 𝐹)) → ({𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ({𝑥} ⊆ 𝑌 ∧ (𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))))
2415, 17, 23mpbir2and 958 . . 3 ((𝜑𝑥 (𝐽 qTop 𝐹)) → {𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)))
2524ralrimiva 2853 . 2 (𝜑 → ∀𝑥 (𝐽 qTop 𝐹){𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)))
26 eqid 2514 . . 3 (𝐽 qTop 𝐹) = (𝐽 qTop 𝐹)
2726ist1 20838 . 2 ((𝐽 qTop 𝐹) ∈ Fre ↔ ((𝐽 qTop 𝐹) ∈ Top ∧ ∀𝑥 (𝐽 qTop 𝐹){𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹))))
289, 25, 27sylanbrc 694 1 (𝜑 → (𝐽 qTop 𝐹) ∈ Fre)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1938  wral 2800  wss 3444  {csn 4028   cuni 4270  ccnv 4931  cima 4935   Fn wfn 5684  ontowfo 5687  cfv 5689  (class class class)co 6426   qTop cqtop 15870  Topctop 20420  TopOnctopon 20421  Clsdccld 20533  Frect1 20824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6723
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-ov 6429  df-oprab 6430  df-mpt2 6431  df-qtop 15875  df-top 20424  df-topon 20426  df-cld 20536  df-t1 20831
This theorem is referenced by: (None)
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