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Theorem qtopss 22011
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 22001, 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 21223 . . . . 5 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑥𝐾) → 𝑥𝑌)
213ad2antl2 1179 . . . 4 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥𝐾) → 𝑥𝑌)
3 cnima 21561 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥𝐾) → (𝐹𝑥) ∈ 𝐽)
433ad2antl1 1178 . . . 4 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥𝐾) → (𝐹𝑥) ∈ 𝐽)
5 simpl1 1184 . . . . . . 7 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
6 cntop1 21536 . . . . . . 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 21214 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
97, 8sylib 219 . . . . 5 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥𝐾) → 𝐽 ∈ (TopOn‘ 𝐽))
10 simpl2 1185 . . . . . . . 8 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥𝐾) → 𝐾 ∈ (TopOn‘𝑌))
11 cnf2 21545 . . . . . . . 8 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹: 𝐽𝑌)
129, 10, 5, 11syl3anc 1364 . . . . . . 7 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥𝐾) → 𝐹: 𝐽𝑌)
1312ffnd 6390 . . . . . 6 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥𝐾) → 𝐹 Fn 𝐽)
14 simpl3 1186 . . . . . 6 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥𝐾) → ran 𝐹 = 𝑌)
15 df-fo 6238 . . . . . 6 (𝐹: 𝐽onto𝑌 ↔ (𝐹 Fn 𝐽 ∧ ran 𝐹 = 𝑌))
1613, 14, 15sylanbrc 583 . . . . 5 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥𝐾) → 𝐹: 𝐽onto𝑌)
17 elqtop3 21999 . . . . 5 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐹: 𝐽onto𝑌) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥𝑌 ∧ (𝐹𝑥) ∈ 𝐽)))
189, 16, 17syl2anc 584 . . . 4 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥𝐾) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥𝑌 ∧ (𝐹𝑥) ∈ 𝐽)))
192, 4, 18mpbir2and 709 . . 3 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥𝐾) → 𝑥 ∈ (𝐽 qTop 𝐹))
2019ex 413 . 2 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) → (𝑥𝐾𝑥 ∈ (𝐽 qTop 𝐹)))
2120ssrdv 3901 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 207  wa 396  w3a 1080   = wceq 1525  wcel 2083  wss 3865   cuni 4751  ccnv 5449  ran crn 5451  cima 5453   Fn wfn 6227  wf 6228  ontowfo 6230  cfv 6232  (class class class)co 7023   qTop cqtop 16609  Topctop 21189  TopOnctopon 21206   Cn ccn 21520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-ov 7026  df-oprab 7027  df-mpo 7028  df-map 8265  df-qtop 16613  df-top 21190  df-topon 21207  df-cn 21523
This theorem is referenced by:  qtoprest  22013  qtopomap  22014  qtopcmap  22015
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