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Theorem qtoptopon 22855
Description: The base set of the quotient topology. (Contributed by Mario Carneiro, 22-Aug-2015.)
Assertion
Ref Expression
qtoptopon ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))

Proof of Theorem qtoptopon
StepHypRef Expression
1 topontop 22062 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2 toponuni 22063 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
3 foeq2 6685 . . . . . 6 (𝑋 = 𝐽 → (𝐹:𝑋onto𝑌𝐹: 𝐽onto𝑌))
42, 3syl 17 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → (𝐹:𝑋onto𝑌𝐹: 𝐽onto𝑌))
54biimpa 477 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → 𝐹: 𝐽onto𝑌)
6 fofn 6690 . . . 4 (𝐹: 𝐽onto𝑌𝐹 Fn 𝐽)
75, 6syl 17 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → 𝐹 Fn 𝐽)
8 eqid 2738 . . . 4 𝐽 = 𝐽
98qtoptop 22851 . . 3 ((𝐽 ∈ Top ∧ 𝐹 Fn 𝐽) → (𝐽 qTop 𝐹) ∈ Top)
101, 7, 9syl2an2r 682 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝐽 qTop 𝐹) ∈ Top)
118qtopuni 22853 . . 3 ((𝐽 ∈ Top ∧ 𝐹: 𝐽onto𝑌) → 𝑌 = (𝐽 qTop 𝐹))
121, 5, 11syl2an2r 682 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → 𝑌 = (𝐽 qTop 𝐹))
13 istopon 22061 . 2 ((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ↔ ((𝐽 qTop 𝐹) ∈ Top ∧ 𝑌 = (𝐽 qTop 𝐹)))
1410, 12, 13sylanbrc 583 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106   cuni 4839   Fn wfn 6428  ontowfo 6431  cfv 6433  (class class class)co 7275   qTop cqtop 17214  Topctop 22042  TopOnctopon 22059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-qtop 17218  df-top 22043  df-topon 22060
This theorem is referenced by:  qtopid  22856  qtopcld  22864  qtopcn  22865  qtopeu  22867  qtoprest  22868  imastps  22872  kqtopon  22878  qtopf1  22967  qtophmeo  22968  qustgplem  23272  qtophaus  31786
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