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Theorem qtopid 23072
Description: A quotient map is a continuous function into its quotient topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
Assertion
Ref Expression
qtopid ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 Fn 𝑋) β†’ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))

Proof of Theorem qtopid
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 simpr 486 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 Fn 𝑋) β†’ 𝐹 Fn 𝑋)
2 dffn4 6767 . . . 4 (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–ontoβ†’ran 𝐹)
31, 2sylib 217 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 Fn 𝑋) β†’ 𝐹:𝑋–ontoβ†’ran 𝐹)
4 fof 6761 . . 3 (𝐹:𝑋–ontoβ†’ran 𝐹 β†’ 𝐹:π‘‹βŸΆran 𝐹)
53, 4syl 17 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 Fn 𝑋) β†’ 𝐹:π‘‹βŸΆran 𝐹)
6 elqtop3 23070 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’ran 𝐹) β†’ (π‘₯ ∈ (𝐽 qTop 𝐹) ↔ (π‘₯ βŠ† ran 𝐹 ∧ (◑𝐹 β€œ π‘₯) ∈ 𝐽)))
73, 6syldan 592 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 Fn 𝑋) β†’ (π‘₯ ∈ (𝐽 qTop 𝐹) ↔ (π‘₯ βŠ† ran 𝐹 ∧ (◑𝐹 β€œ π‘₯) ∈ 𝐽)))
87simplbda 501 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 Fn 𝑋) ∧ π‘₯ ∈ (𝐽 qTop 𝐹)) β†’ (◑𝐹 β€œ π‘₯) ∈ 𝐽)
98ralrimiva 3144 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 Fn 𝑋) β†’ βˆ€π‘₯ ∈ (𝐽 qTop 𝐹)(◑𝐹 β€œ π‘₯) ∈ 𝐽)
10 qtoptopon 23071 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’ran 𝐹) β†’ (𝐽 qTop 𝐹) ∈ (TopOnβ€˜ran 𝐹))
113, 10syldan 592 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 Fn 𝑋) β†’ (𝐽 qTop 𝐹) ∈ (TopOnβ€˜ran 𝐹))
12 iscn 22602 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝐽 qTop 𝐹) ∈ (TopOnβ€˜ran 𝐹)) β†’ (𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)) ↔ (𝐹:π‘‹βŸΆran 𝐹 ∧ βˆ€π‘₯ ∈ (𝐽 qTop 𝐹)(◑𝐹 β€œ π‘₯) ∈ 𝐽)))
1311, 12syldan 592 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 Fn 𝑋) β†’ (𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)) ↔ (𝐹:π‘‹βŸΆran 𝐹 ∧ βˆ€π‘₯ ∈ (𝐽 qTop 𝐹)(◑𝐹 β€œ π‘₯) ∈ 𝐽)))
145, 9, 13mpbir2and 712 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 Fn 𝑋) β†’ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∈ wcel 2107  βˆ€wral 3065   βŠ† wss 3915  β—‘ccnv 5637  ran crn 5639   β€œ cima 5641   Fn wfn 6496  βŸΆwf 6497  β€“ontoβ†’wfo 6499  β€˜cfv 6501  (class class class)co 7362   qTop cqtop 17392  TopOnctopon 22275   Cn ccn 22591
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 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
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 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-map 8774  df-qtop 17396  df-top 22259  df-topon 22276  df-cn 22594
This theorem is referenced by:  qtopcmplem  23074  qtopkgen  23077  qtoprest  23084  kqid  23095  qtopf1  23183  qtophmeo  23184  qustgplem  23488  circcn  32459
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