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Theorem qtopss 22327
 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 22317, 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 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) → 𝐾 ⊆ (𝐽 qTop 𝐹))

Proof of Theorem qtopss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 toponss 21539 . . . . 5 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑥𝐾) → 𝑥𝑌)
213ad2antl2 1183 . . . 4 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥𝐾) → 𝑥𝑌)
3 cnima 21877 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥𝐾) → (𝐹𝑥) ∈ 𝐽)
433ad2antl1 1182 . . . 4 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥𝐾) → (𝐹𝑥) ∈ 𝐽)
5 simpl1 1188 . . . . . . 7 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
6 cntop1 21852 . . . . . . 7 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
75, 6syl 17 . . . . . 6 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥𝐾) → 𝐽 ∈ Top)
8 toptopon2 21530 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
97, 8sylib 221 . . . . 5 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥𝐾) → 𝐽 ∈ (TopOn‘ 𝐽))
10 simpl2 1189 . . . . . . . 8 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥𝐾) → 𝐾 ∈ (TopOn‘𝑌))
11 cnf2 21861 . . . . . . . 8 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹: 𝐽𝑌)
129, 10, 5, 11syl3anc 1368 . . . . . . 7 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥𝐾) → 𝐹: 𝐽𝑌)
1312ffnd 6488 . . . . . 6 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥𝐾) → 𝐹 Fn 𝐽)
14 simpl3 1190 . . . . . 6 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥𝐾) → ran 𝐹 = 𝑌)
15 df-fo 6330 . . . . . 6 (𝐹: 𝐽onto𝑌 ↔ (𝐹 Fn 𝐽 ∧ ran 𝐹 = 𝑌))
1613, 14, 15sylanbrc 586 . . . . 5 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥𝐾) → 𝐹: 𝐽onto𝑌)
17 elqtop3 22315 . . . . 5 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐹: 𝐽onto𝑌) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥𝑌 ∧ (𝐹𝑥) ∈ 𝐽)))
189, 16, 17syl2anc 587 . . . 4 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥𝐾) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥𝑌 ∧ (𝐹𝑥) ∈ 𝐽)))
192, 4, 18mpbir2and 712 . . 3 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥𝐾) → 𝑥 ∈ (𝐽 qTop 𝐹))
2019ex 416 . 2 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) → (𝑥𝐾𝑥 ∈ (𝐽 qTop 𝐹)))
2120ssrdv 3921 1 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) → 𝐾 ⊆ (𝐽 qTop 𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ⊆ wss 3881  ∪ cuni 4800  ◡ccnv 5518  ran crn 5520   “ cima 5522   Fn wfn 6319  ⟶wf 6320  –onto→wfo 6322  ‘cfv 6324  (class class class)co 7135   qTop cqtop 16770  Topctop 21505  TopOnctopon 21522   Cn ccn 21836 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-map 8393  df-qtop 16774  df-top 21506  df-topon 21523  df-cn 21839 This theorem is referenced by:  qtoprest  22329  qtopomap  22330  qtopcmap  22331
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