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Theorem qtopss 23563
Description: A surjective continuous function from 𝐽 to 𝐾 induces a topology 𝐽 qTop 𝐹 on the base set of 𝐾. This topology is in general finer than 𝐾. Together with qtopid 23553, this implies that 𝐽 qTop 𝐹 is the finest topology making 𝐹 continuous, i.e. the final topology with respect to the family {𝐹}. (Contributed by Mario Carneiro, 24-Mar-2015.)
Assertion
Ref Expression
qtopss ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 = π‘Œ) β†’ 𝐾 βŠ† (𝐽 qTop 𝐹))

Proof of Theorem qtopss
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 toponss 22773 . . . . 5 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ π‘₯ βŠ† π‘Œ)
213ad2antl2 1183 . . . 4 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 = π‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ π‘₯ βŠ† π‘Œ)
3 cnima 23113 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ π‘₯ ∈ 𝐾) β†’ (◑𝐹 β€œ π‘₯) ∈ 𝐽)
433ad2antl1 1182 . . . 4 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 = π‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ (◑𝐹 β€œ π‘₯) ∈ 𝐽)
5 simpl1 1188 . . . . . . 7 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 = π‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ 𝐹 ∈ (𝐽 Cn 𝐾))
6 cntop1 23088 . . . . . . 7 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐽 ∈ Top)
75, 6syl 17 . . . . . 6 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 = π‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ 𝐽 ∈ Top)
8 toptopon2 22764 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
97, 8sylib 217 . . . . 5 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 = π‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
10 simpl2 1189 . . . . . . . 8 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 = π‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
11 cnf2 23097 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐹:βˆͺ π½βŸΆπ‘Œ)
129, 10, 5, 11syl3anc 1368 . . . . . . 7 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 = π‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ 𝐹:βˆͺ π½βŸΆπ‘Œ)
1312ffnd 6709 . . . . . 6 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 = π‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ 𝐹 Fn βˆͺ 𝐽)
14 simpl3 1190 . . . . . 6 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 = π‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ ran 𝐹 = π‘Œ)
15 df-fo 6540 . . . . . 6 (𝐹:βˆͺ 𝐽–ontoβ†’π‘Œ ↔ (𝐹 Fn βˆͺ 𝐽 ∧ ran 𝐹 = π‘Œ))
1613, 14, 15sylanbrc 582 . . . . 5 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 = π‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ 𝐹:βˆͺ 𝐽–ontoβ†’π‘Œ)
17 elqtop3 23551 . . . . 5 ((𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽) ∧ 𝐹:βˆͺ 𝐽–ontoβ†’π‘Œ) β†’ (π‘₯ ∈ (𝐽 qTop 𝐹) ↔ (π‘₯ βŠ† π‘Œ ∧ (◑𝐹 β€œ π‘₯) ∈ 𝐽)))
189, 16, 17syl2anc 583 . . . 4 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 = π‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ (π‘₯ ∈ (𝐽 qTop 𝐹) ↔ (π‘₯ βŠ† π‘Œ ∧ (◑𝐹 β€œ π‘₯) ∈ 𝐽)))
192, 4, 18mpbir2and 710 . . 3 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 = π‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ π‘₯ ∈ (𝐽 qTop 𝐹))
2019ex 412 . 2 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 = π‘Œ) β†’ (π‘₯ ∈ 𝐾 β†’ π‘₯ ∈ (𝐽 qTop 𝐹)))
2120ssrdv 3981 1 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 = π‘Œ) β†’ 𝐾 βŠ† (𝐽 qTop 𝐹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   βŠ† wss 3941  βˆͺ cuni 4900  β—‘ccnv 5666  ran crn 5668   β€œ cima 5670   Fn wfn 6529  βŸΆwf 6530  β€“ontoβ†’wfo 6532  β€˜cfv 6534  (class class class)co 7402   qTop cqtop 17454  Topctop 22739  TopOnctopon 22756   Cn ccn 23072
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 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
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 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-map 8819  df-qtop 17458  df-top 22740  df-topon 22757  df-cn 23075
This theorem is referenced by:  qtoprest  23565  qtopomap  23566  qtopcmap  23567
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