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Theorem qtopid 23560
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 484 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 Fn 𝑋) β†’ 𝐹 Fn 𝑋)
2 dffn4 6804 . . . 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 6798 . . 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 23558 . . . . 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 590 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 Fn 𝑋) β†’ (π‘₯ ∈ (𝐽 qTop 𝐹) ↔ (π‘₯ βŠ† ran 𝐹 ∧ (◑𝐹 β€œ π‘₯) ∈ 𝐽)))
87simplbda 499 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 Fn 𝑋) ∧ π‘₯ ∈ (𝐽 qTop 𝐹)) β†’ (◑𝐹 β€œ π‘₯) ∈ 𝐽)
98ralrimiva 3140 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 Fn 𝑋) β†’ βˆ€π‘₯ ∈ (𝐽 qTop 𝐹)(◑𝐹 β€œ π‘₯) ∈ 𝐽)
10 qtoptopon 23559 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’ran 𝐹) β†’ (𝐽 qTop 𝐹) ∈ (TopOnβ€˜ran 𝐹))
113, 10syldan 590 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 Fn 𝑋) β†’ (𝐽 qTop 𝐹) ∈ (TopOnβ€˜ran 𝐹))
12 iscn 23090 . . 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 590 . 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 710 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 395   ∈ wcel 2098  βˆ€wral 3055   βŠ† wss 3943  β—‘ccnv 5668  ran crn 5670   β€œ cima 5672   Fn wfn 6531  βŸΆwf 6532  β€“ontoβ†’wfo 6534  β€˜cfv 6536  (class class class)co 7404   qTop cqtop 17456  TopOnctopon 22763   Cn ccn 23079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-map 8821  df-qtop 17460  df-top 22747  df-topon 22764  df-cn 23082
This theorem is referenced by:  qtopcmplem  23562  qtopkgen  23565  qtoprest  23572  kqid  23583  qtopf1  23671  qtophmeo  23672  qustgplem  23976  circcn  33348
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