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Theorem qtoptop 23054
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 484 . 2 ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → 𝐽 ∈ Top)
2 id 22 . . 3 (𝐹 Fn 𝑋𝐹 Fn 𝑋)
3 qtoptop.1 . . . 4 𝑋 = 𝐽
43topopn 22258 . . 3 (𝐽 ∈ Top → 𝑋𝐽)
5 fnex 7168 . . 3 ((𝐹 Fn 𝑋𝑋𝐽) → 𝐹 ∈ V)
62, 4, 5syl2anr 598 . 2 ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ V)
7 fnfun 6603 . . 3 (𝐹 Fn 𝑋 → Fun 𝐹)
87adantl 483 . 2 ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → Fun 𝐹)
9 qtoptop2 23053 . 2 ((𝐽 ∈ Top ∧ 𝐹 ∈ V ∧ Fun 𝐹) → (𝐽 qTop 𝐹) ∈ Top)
101, 6, 8, 9syl3anc 1372 1 ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  Vcvv 3446   cuni 4866  Fun wfun 6491   Fn wfn 6492  (class class class)co 7358   qTop cqtop 17386  Topctop 22245
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-qtop 17390  df-top 22246
This theorem is referenced by:  qtoptopon  23058  qtopkgen  23064  qtopt1  32419  qtophaus  32420
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