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Theorem qtopid 22007
Description: A quotient map is a continuous function into its quotient topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
Assertion
Ref Expression
qtopid ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))

Proof of Theorem qtopid
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simpr 477 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 Fn 𝑋)
2 dffn4 6419 . . . 4 (𝐹 Fn 𝑋𝐹:𝑋onto→ran 𝐹)
31, 2sylib 210 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹:𝑋onto→ran 𝐹)
4 fof 6413 . . 3 (𝐹:𝑋onto→ran 𝐹𝐹:𝑋⟶ran 𝐹)
53, 4syl 17 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹:𝑋⟶ran 𝐹)
6 elqtop3 22005 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto→ran 𝐹) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ ran 𝐹 ∧ (𝐹𝑥) ∈ 𝐽)))
73, 6syldan 582 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ ran 𝐹 ∧ (𝐹𝑥) ∈ 𝐽)))
87simplbda 492 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) ∧ 𝑥 ∈ (𝐽 qTop 𝐹)) → (𝐹𝑥) ∈ 𝐽)
98ralrimiva 3126 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → ∀𝑥 ∈ (𝐽 qTop 𝐹)(𝐹𝑥) ∈ 𝐽)
10 qtoptopon 22006 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto→ran 𝐹) → (𝐽 qTop 𝐹) ∈ (TopOn‘ran 𝐹))
113, 10syldan 582 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ (TopOn‘ran 𝐹))
12 iscn 21537 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐽 qTop 𝐹) ∈ (TopOn‘ran 𝐹)) → (𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)) ↔ (𝐹:𝑋⟶ran 𝐹 ∧ ∀𝑥 ∈ (𝐽 qTop 𝐹)(𝐹𝑥) ∈ 𝐽)))
1311, 12syldan 582 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → (𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)) ↔ (𝐹:𝑋⟶ran 𝐹 ∧ ∀𝑥 ∈ (𝐽 qTop 𝐹)(𝐹𝑥) ∈ 𝐽)))
145, 9, 13mpbir2and 700 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  wcel 2048  wral 3082  wss 3825  ccnv 5399  ran crn 5401  cima 5403   Fn wfn 6177  wf 6178  ontowfo 6180  cfv 6182  (class class class)co 6970   qTop cqtop 16622  TopOnctopon 21212   Cn ccn 21526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-ov 6973  df-oprab 6974  df-mpo 6975  df-map 8200  df-qtop 16626  df-top 21196  df-topon 21213  df-cn 21529
This theorem is referenced by:  qtopcmplem  22009  qtopkgen  22012  qtoprest  22019  kqid  22030  qtopf1  22118  qtophmeo  22119  qustgplem  22422  circcn  30703
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