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Theorem qtopid 23627
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 483 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 Fn 𝑋) β†’ 𝐹 Fn 𝑋)
2 dffn4 6820 . . . 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 6814 . . 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 23625 . . . . 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 589 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 Fn 𝑋) β†’ (π‘₯ ∈ (𝐽 qTop 𝐹) ↔ (π‘₯ βŠ† ran 𝐹 ∧ (◑𝐹 β€œ π‘₯) ∈ 𝐽)))
87simplbda 498 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 Fn 𝑋) ∧ π‘₯ ∈ (𝐽 qTop 𝐹)) β†’ (◑𝐹 β€œ π‘₯) ∈ 𝐽)
98ralrimiva 3142 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 Fn 𝑋) β†’ βˆ€π‘₯ ∈ (𝐽 qTop 𝐹)(◑𝐹 β€œ π‘₯) ∈ 𝐽)
10 qtoptopon 23626 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’ran 𝐹) β†’ (𝐽 qTop 𝐹) ∈ (TopOnβ€˜ran 𝐹))
113, 10syldan 589 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 Fn 𝑋) β†’ (𝐽 qTop 𝐹) ∈ (TopOnβ€˜ran 𝐹))
12 iscn 23157 . . 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 589 . 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 711 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 394   ∈ wcel 2098  βˆ€wral 3057   βŠ† wss 3947  β—‘ccnv 5679  ran crn 5681   β€œ cima 5683   Fn wfn 6546  βŸΆwf 6547  β€“ontoβ†’wfo 6549  β€˜cfv 6551  (class class class)co 7424   qTop cqtop 17490  TopOnctopon 22830   Cn ccn 23146
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 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429  df-map 8851  df-qtop 17494  df-top 22814  df-topon 22831  df-cn 23149
This theorem is referenced by:  qtopcmplem  23629  qtopkgen  23632  qtoprest  23639  kqid  23650  qtopf1  23738  qtophmeo  23739  qustgplem  24043  circcn  33444
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