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Theorem qtopt1 30740
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 21642 . . . 4 (𝐽 ∈ Fre → 𝐽 ∈ Top)
31, 2syl 17 . . 3 (𝜑𝐽 ∈ Top)
4 qtopt1.2 . . . 4 (𝜑𝐹:𝑋onto𝑌)
5 fofn 6421 . . . 4 (𝐹:𝑋onto𝑌𝐹 Fn 𝑋)
64, 5syl 17 . . 3 (𝜑𝐹 Fn 𝑋)
7 qtopt1.x . . . 4 𝑋 = 𝐽
87qtoptop 22012 . . 3 ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Top)
93, 6, 8syl2anc 576 . 2 (𝜑 → (𝐽 qTop 𝐹) ∈ Top)
10 simpr 477 . . . . . 6 ((𝜑𝑥 (𝐽 qTop 𝐹)) → 𝑥 (𝐽 qTop 𝐹))
117qtopuni 22014 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐹:𝑋onto𝑌) → 𝑌 = (𝐽 qTop 𝐹))
123, 4, 11syl2anc 576 . . . . . . 7 (𝜑𝑌 = (𝐽 qTop 𝐹))
1312adantr 473 . . . . . 6 ((𝜑𝑥 (𝐽 qTop 𝐹)) → 𝑌 = (𝐽 qTop 𝐹))
1410, 13eleqtrrd 2870 . . . . 5 ((𝜑𝑥 (𝐽 qTop 𝐹)) → 𝑥𝑌)
1514snssd 4616 . . . 4 ((𝜑𝑥 (𝐽 qTop 𝐹)) → {𝑥} ⊆ 𝑌)
16 qtopt1.3 . . . . 5 ((𝜑𝑥𝑌) → (𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))
1714, 16syldan 582 . . . 4 ((𝜑𝑥 (𝐽 qTop 𝐹)) → (𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))
183, 7jctir 513 . . . . . . 7 (𝜑 → (𝐽 ∈ Top ∧ 𝑋 = 𝐽))
19 istopon 21224 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝐽))
2018, 19sylibr 226 . . . . . 6 (𝜑𝐽 ∈ (TopOn‘𝑋))
21 qtopcld 22025 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → ({𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ({𝑥} ⊆ 𝑌 ∧ (𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))))
2220, 4, 21syl2anc 576 . . . . 5 (𝜑 → ({𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ({𝑥} ⊆ 𝑌 ∧ (𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))))
2322adantr 473 . . . 4 ((𝜑𝑥 (𝐽 qTop 𝐹)) → ({𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ({𝑥} ⊆ 𝑌 ∧ (𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))))
2415, 17, 23mpbir2and 700 . . 3 ((𝜑𝑥 (𝐽 qTop 𝐹)) → {𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)))
2524ralrimiva 3133 . 2 (𝜑 → ∀𝑥 (𝐽 qTop 𝐹){𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)))
26 eqid 2779 . . 3 (𝐽 qTop 𝐹) = (𝐽 qTop 𝐹)
2726ist1 21633 . 2 ((𝐽 qTop 𝐹) ∈ Fre ↔ ((𝐽 qTop 𝐹) ∈ Top ∧ ∀𝑥 (𝐽 qTop 𝐹){𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹))))
289, 25, 27sylanbrc 575 1 (𝜑 → (𝐽 qTop 𝐹) ∈ Fre)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1507  wcel 2050  wral 3089  wss 3830  {csn 4441   cuni 4712  ccnv 5406  cima 5410   Fn wfn 6183  ontowfo 6186  cfv 6188  (class class class)co 6976   qTop cqtop 16632  Topctop 21205  TopOnctopon 21222  Clsdccld 21328  Frect1 21619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-ov 6979  df-oprab 6980  df-mpo 6981  df-qtop 16636  df-top 21206  df-topon 21223  df-cld 21331  df-t1 21626
This theorem is referenced by: (None)
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