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Theorem qtoptop 23724
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 22928 . . 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 6669 . . 3 (𝐹 Fn 𝑋 → Fun 𝐹)
87adantl 481 . 2 ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → Fun 𝐹)
9 qtoptop2 23723 . 2 ((𝐽 ∈ Top ∧ 𝐹 ∈ V ∧ Fun 𝐹) → (𝐽 qTop 𝐹) ∈ Top)
101, 6, 8, 9syl3anc 1370 1 ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478   cuni 4912  Fun wfun 6557   Fn wfn 6558  (class class class)co 7431   qTop cqtop 17550  Topctop 22915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-qtop 17554  df-top 22916
This theorem is referenced by:  qtoptopon  23728  qtopkgen  23734  qtopt1  33796  qtophaus  33797
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