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Theorem qtoptop 23526
Description: The quotient topology is a topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
Hypothesis
Ref Expression
qtoptop.1 𝑋 = 𝐽
Assertion
Ref Expression
qtoptop ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Top)

Proof of Theorem qtoptop
StepHypRef Expression
1 simpl 482 . 2 ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → 𝐽 ∈ Top)
2 id 22 . . 3 (𝐹 Fn 𝑋𝐹 Fn 𝑋)
3 qtoptop.1 . . . 4 𝑋 = 𝐽
43topopn 22730 . . 3 (𝐽 ∈ Top → 𝑋𝐽)
5 fnex 7210 . . 3 ((𝐹 Fn 𝑋𝑋𝐽) → 𝐹 ∈ V)
62, 4, 5syl2anr 596 . 2 ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ V)
7 fnfun 6639 . . 3 (𝐹 Fn 𝑋 → Fun 𝐹)
87adantl 481 . 2 ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → Fun 𝐹)
9 qtoptop2 23525 . 2 ((𝐽 ∈ Top ∧ 𝐹 ∈ V ∧ Fun 𝐹) → (𝐽 qTop 𝐹) ∈ Top)
101, 6, 8, 9syl3anc 1368 1 ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  Vcvv 3466   cuni 4899  Fun wfun 6527   Fn wfn 6528  (class class class)co 7401   qTop cqtop 17448  Topctop 22717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-qtop 17452  df-top 22718
This theorem is referenced by:  qtoptopon  23530  qtopkgen  23536  qtopt1  33304  qtophaus  33305
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