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Theorem qtoptop 23708
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 22912 . . 3 (𝐽 ∈ Top → 𝑋𝐽)
5 fnex 7237 . . 3 ((𝐹 Fn 𝑋𝑋𝐽) → 𝐹 ∈ V)
62, 4, 5syl2anr 597 . 2 ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ V)
7 fnfun 6668 . . 3 (𝐹 Fn 𝑋 → Fun 𝐹)
87adantl 481 . 2 ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → Fun 𝐹)
9 qtoptop2 23707 . 2 ((𝐽 ∈ Top ∧ 𝐹 ∈ V ∧ Fun 𝐹) → (𝐽 qTop 𝐹) ∈ Top)
101, 6, 8, 9syl3anc 1373 1 ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480   cuni 4907  Fun wfun 6555   Fn wfn 6556  (class class class)co 7431   qTop cqtop 17548  Topctop 22899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-qtop 17552  df-top 22900
This theorem is referenced by:  qtoptopon  23712  qtopkgen  23718  qtopt1  33834  qtophaus  33835
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