Theorem qtopss 23763

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 23753, 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.

Step	Hyp	Ref	Expression
1		toponss 22975	. . . . 5 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑥 ∈ 𝐾) → 𝑥 ⊆ 𝑌)
2	1	3ad2antl2 1199	. . . 4 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → 𝑥 ⊆ 𝑌)
3		cnima 23313	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 ∈ 𝐾) → (◡𝐹 “ 𝑥) ∈ 𝐽)
4	3	3ad2antl1 1198	. . . 4 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → (◡𝐹 “ 𝑥) ∈ 𝐽)
5		simpl1 1204	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
6		cntop1 23288	. . . . . . 7 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
7	5, 6	syl 17	. . . . . 6 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → 𝐽 ∈ Top)
8		toptopon2 22966	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
9	7, 8	sylib 220	. . . . 5 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → 𝐽 ∈ (TopOn‘∪ 𝐽))
10		simpl2 1205	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → 𝐾 ∈ (TopOn‘𝑌))
11		cnf2 23297	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:∪ 𝐽⟶𝑌)
12	9, 10, 5, 11	syl3anc 1389	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → 𝐹:∪ 𝐽⟶𝑌)
13	12	ffnd 6687	. . . . . 6 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → 𝐹 Fn ∪ 𝐽)
14		simpl3 1206	. . . . . 6 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → ran 𝐹 = 𝑌)
15		df-fo 6522	. . . . . 6 ⊢ (𝐹:∪ 𝐽–onto→𝑌 ↔ (𝐹 Fn ∪ 𝐽 ∧ ran 𝐹 = 𝑌))
16	13, 14, 15	sylanbrc 592	. . . . 5 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → 𝐹:∪ 𝐽–onto→𝑌)
17		elqtop3 23751	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐹:∪ 𝐽–onto→𝑌) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽)))
18	9, 16, 17	syl2anc 593	. . . 4 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽)))
19	2, 4, 18	mpbir2and 723	. . 3 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) ∧ 𝑥 ∈ 𝐾) → 𝑥 ∈ (𝐽 qTop 𝐹))
20	19	ex 416	. 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) → (𝑥 ∈ 𝐾 → 𝑥 ∈ (𝐽 qTop 𝐹)))
21	20	ssrdv 3940	1 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹 = 𝑌) → 𝐾 ⊆ (𝐽 qTop 𝐹))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ⊆ wss 3902  ∪ cuni 4862  ^◡ccnv 5642  ran crn 5644   “ cima 5646   Fn wfn 6511  ⟶wf 6512  –onto→wfo 6514  ‘cfv 6516  (class class class)co 7391   qTop cqtop 17524  Topctop 22941  TopOnctopon 22958   Cn ccn 23272

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7394  df-oprab 7395  df-mpo 7396  df-map 8804  df-qtop 17528  df-top 22942  df-topon 22959  df-cn 23275

This theorem is referenced by:  qtoprest  23765  qtopomap  23766  qtopcmap  23767