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Theorem qtopconn 23532
Description: A quotient of a connected space is connected. (Contributed by Mario Carneiro, 24-Mar-2015.)
Hypothesis
Ref Expression
qtopcmp.1 𝑋 = 𝐽
Assertion
Ref Expression
qtopconn ((𝐽 ∈ Conn ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Conn)

Proof of Theorem qtopconn
StepHypRef Expression
1 qtopcmp.1 . 2 𝑋 = 𝐽
2 conntop 23240 . 2 (𝐽 ∈ Conn → 𝐽 ∈ Top)
3 eqid 2731 . . 3 (𝐽 qTop 𝐹) = (𝐽 qTop 𝐹)
43cnconn 23245 . 2 ((𝐽 ∈ Conn ∧ 𝐹:𝑋onto (𝐽 qTop 𝐹) ∧ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹))) → (𝐽 qTop 𝐹) ∈ Conn)
51, 2, 4qtopcmplem 23530 1 ((𝐽 ∈ Conn ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Conn)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105   cuni 4908   Fn wfn 6538  (class class class)co 7412   qTop cqtop 17456  Conncconn 23234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8828  df-qtop 17460  df-top 22715  df-topon 22732  df-cld 22842  df-cn 23050  df-conn 23235
This theorem is referenced by: (None)
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