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Theorem qtopss 23089
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 23079, 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 22299 . . . . 5 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ π‘₯ βŠ† π‘Œ)
213ad2antl2 1187 . . . 4 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 = π‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ π‘₯ βŠ† π‘Œ)
3 cnima 22639 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ π‘₯ ∈ 𝐾) β†’ (◑𝐹 β€œ π‘₯) ∈ 𝐽)
433ad2antl1 1186 . . . 4 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 = π‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ (◑𝐹 β€œ π‘₯) ∈ 𝐽)
5 simpl1 1192 . . . . . . 7 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 = π‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ 𝐹 ∈ (𝐽 Cn 𝐾))
6 cntop1 22614 . . . . . . 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 22290 . . . . . 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 1193 . . . . . . . 8 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 = π‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
11 cnf2 22623 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐹:βˆͺ π½βŸΆπ‘Œ)
129, 10, 5, 11syl3anc 1372 . . . . . . 7 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 = π‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ 𝐹:βˆͺ π½βŸΆπ‘Œ)
1312ffnd 6673 . . . . . 6 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 = π‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ 𝐹 Fn βˆͺ 𝐽)
14 simpl3 1194 . . . . . 6 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 = π‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ ran 𝐹 = π‘Œ)
15 df-fo 6506 . . . . . 6 (𝐹:βˆͺ 𝐽–ontoβ†’π‘Œ ↔ (𝐹 Fn βˆͺ 𝐽 ∧ ran 𝐹 = π‘Œ))
1613, 14, 15sylanbrc 584 . . . . 5 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 = π‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ 𝐹:βˆͺ 𝐽–ontoβ†’π‘Œ)
17 elqtop3 23077 . . . . 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 585 . . . 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 712 . . 3 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 = π‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ π‘₯ ∈ (𝐽 qTop 𝐹))
2019ex 414 . 2 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 = π‘Œ) β†’ (π‘₯ ∈ 𝐾 β†’ π‘₯ ∈ (𝐽 qTop 𝐹)))
2120ssrdv 3954 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   βŠ† wss 3914  βˆͺ cuni 4869  β—‘ccnv 5636  ran crn 5638   β€œ cima 5640   Fn wfn 6495  βŸΆwf 6496  β€“ontoβ†’wfo 6498  β€˜cfv 6500  (class class class)co 7361   qTop cqtop 17393  Topctop 22265  TopOnctopon 22282   Cn ccn 22598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-map 8773  df-qtop 17397  df-top 22266  df-topon 22283  df-cn 22601
This theorem is referenced by:  qtoprest  23091  qtopomap  23092  qtopcmap  23093
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