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Theorem qtopf1 23824
Description: If a quotient map is injective, then it is a homeomorphism. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypotheses
Ref Expression
qtopf1.1 (𝜑𝐽 ∈ (TopOn‘𝑋))
qtopf1.2 (𝜑𝐹:𝑋1-1𝑌)
Assertion
Ref Expression
qtopf1 (𝜑𝐹 ∈ (𝐽Homeo(𝐽 qTop 𝐹)))

Proof of Theorem qtopf1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 qtopf1.1 . . 3 (𝜑𝐽 ∈ (TopOn‘𝑋))
2 qtopf1.2 . . . 4 (𝜑𝐹:𝑋1-1𝑌)
3 f1fn 6805 . . . 4 (𝐹:𝑋1-1𝑌𝐹 Fn 𝑋)
42, 3syl 17 . . 3 (𝜑𝐹 Fn 𝑋)
5 qtopid 23713 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
61, 4, 5syl2anc 584 . 2 (𝜑𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
7 f1f1orn 6859 . . . 4 (𝐹:𝑋1-1𝑌𝐹:𝑋1-1-onto→ran 𝐹)
8 f1ocnv 6860 . . . 4 (𝐹:𝑋1-1-onto→ran 𝐹𝐹:ran 𝐹1-1-onto𝑋)
9 f1of 6848 . . . 4 (𝐹:ran 𝐹1-1-onto𝑋𝐹:ran 𝐹𝑋)
102, 7, 8, 94syl 19 . . 3 (𝜑𝐹:ran 𝐹𝑋)
11 imacnvcnv 6226 . . . . 5 (𝐹𝑥) = (𝐹𝑥)
12 imassrn 6089 . . . . . . 7 (𝐹𝑥) ⊆ ran 𝐹
1312a1i 11 . . . . . 6 ((𝜑𝑥𝐽) → (𝐹𝑥) ⊆ ran 𝐹)
142adantr 480 . . . . . . . 8 ((𝜑𝑥𝐽) → 𝐹:𝑋1-1𝑌)
15 toponss 22933 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝐽) → 𝑥𝑋)
161, 15sylan 580 . . . . . . . 8 ((𝜑𝑥𝐽) → 𝑥𝑋)
17 f1imacnv 6864 . . . . . . . 8 ((𝐹:𝑋1-1𝑌𝑥𝑋) → (𝐹 “ (𝐹𝑥)) = 𝑥)
1814, 16, 17syl2anc 584 . . . . . . 7 ((𝜑𝑥𝐽) → (𝐹 “ (𝐹𝑥)) = 𝑥)
19 simpr 484 . . . . . . 7 ((𝜑𝑥𝐽) → 𝑥𝐽)
2018, 19eqeltrd 2841 . . . . . 6 ((𝜑𝑥𝐽) → (𝐹 “ (𝐹𝑥)) ∈ 𝐽)
21 dffn4 6826 . . . . . . . . 9 (𝐹 Fn 𝑋𝐹:𝑋onto→ran 𝐹)
224, 21sylib 218 . . . . . . . 8 (𝜑𝐹:𝑋onto→ran 𝐹)
23 elqtop3 23711 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto→ran 𝐹) → ((𝐹𝑥) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹𝑥) ⊆ ran 𝐹 ∧ (𝐹 “ (𝐹𝑥)) ∈ 𝐽)))
241, 22, 23syl2anc 584 . . . . . . 7 (𝜑 → ((𝐹𝑥) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹𝑥) ⊆ ran 𝐹 ∧ (𝐹 “ (𝐹𝑥)) ∈ 𝐽)))
2524adantr 480 . . . . . 6 ((𝜑𝑥𝐽) → ((𝐹𝑥) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹𝑥) ⊆ ran 𝐹 ∧ (𝐹 “ (𝐹𝑥)) ∈ 𝐽)))
2613, 20, 25mpbir2and 713 . . . . 5 ((𝜑𝑥𝐽) → (𝐹𝑥) ∈ (𝐽 qTop 𝐹))
2711, 26eqeltrid 2845 . . . 4 ((𝜑𝑥𝐽) → (𝐹𝑥) ∈ (𝐽 qTop 𝐹))
2827ralrimiva 3146 . . 3 (𝜑 → ∀𝑥𝐽 (𝐹𝑥) ∈ (𝐽 qTop 𝐹))
29 qtoptopon 23712 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto→ran 𝐹) → (𝐽 qTop 𝐹) ∈ (TopOn‘ran 𝐹))
301, 22, 29syl2anc 584 . . . 4 (𝜑 → (𝐽 qTop 𝐹) ∈ (TopOn‘ran 𝐹))
31 iscn 23243 . . . 4 (((𝐽 qTop 𝐹) ∈ (TopOn‘ran 𝐹) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐹 ∈ ((𝐽 qTop 𝐹) Cn 𝐽) ↔ (𝐹:ran 𝐹𝑋 ∧ ∀𝑥𝐽 (𝐹𝑥) ∈ (𝐽 qTop 𝐹))))
3230, 1, 31syl2anc 584 . . 3 (𝜑 → (𝐹 ∈ ((𝐽 qTop 𝐹) Cn 𝐽) ↔ (𝐹:ran 𝐹𝑋 ∧ ∀𝑥𝐽 (𝐹𝑥) ∈ (𝐽 qTop 𝐹))))
3310, 28, 32mpbir2and 713 . 2 (𝜑𝐹 ∈ ((𝐽 qTop 𝐹) Cn 𝐽))
34 ishmeo 23767 . 2 (𝐹 ∈ (𝐽Homeo(𝐽 qTop 𝐹)) ↔ (𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)) ∧ 𝐹 ∈ ((𝐽 qTop 𝐹) Cn 𝐽)))
356, 33, 34sylanbrc 583 1 (𝜑𝐹 ∈ (𝐽Homeo(𝐽 qTop 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wss 3951  ccnv 5684  ran crn 5686  cima 5688   Fn wfn 6556  wf 6557  1-1wf1 6558  ontowfo 6559  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431   qTop cqtop 17548  TopOnctopon 22916   Cn ccn 23232  Homeochmeo 23761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-qtop 17552  df-top 22900  df-topon 22917  df-cn 23235  df-hmeo 23763
This theorem is referenced by:  t0kq  23826
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