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Theorem qtopf1 21841
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 6263 . . . 4 (𝐹:𝑋1-1𝑌𝐹 Fn 𝑋)
42, 3syl 17 . . 3 (𝜑𝐹 Fn 𝑋)
5 qtopid 21730 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
61, 4, 5syl2anc 696 . 2 (𝜑𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
7 f1f1orn 6310 . . . 4 (𝐹:𝑋1-1𝑌𝐹:𝑋1-1-onto→ran 𝐹)
8 f1ocnv 6311 . . . 4 (𝐹:𝑋1-1-onto→ran 𝐹𝐹:ran 𝐹1-1-onto𝑋)
9 f1of 6299 . . . 4 (𝐹:ran 𝐹1-1-onto𝑋𝐹:ran 𝐹𝑋)
102, 7, 8, 94syl 19 . . 3 (𝜑𝐹:ran 𝐹𝑋)
11 imacnvcnv 5757 . . . . 5 (𝐹𝑥) = (𝐹𝑥)
12 imassrn 5635 . . . . . . 7 (𝐹𝑥) ⊆ ran 𝐹
1312a1i 11 . . . . . 6 ((𝜑𝑥𝐽) → (𝐹𝑥) ⊆ ran 𝐹)
142adantr 472 . . . . . . . 8 ((𝜑𝑥𝐽) → 𝐹:𝑋1-1𝑌)
15 toponss 20953 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝐽) → 𝑥𝑋)
161, 15sylan 489 . . . . . . . 8 ((𝜑𝑥𝐽) → 𝑥𝑋)
17 f1imacnv 6315 . . . . . . . 8 ((𝐹:𝑋1-1𝑌𝑥𝑋) → (𝐹 “ (𝐹𝑥)) = 𝑥)
1814, 16, 17syl2anc 696 . . . . . . 7 ((𝜑𝑥𝐽) → (𝐹 “ (𝐹𝑥)) = 𝑥)
19 simpr 479 . . . . . . 7 ((𝜑𝑥𝐽) → 𝑥𝐽)
2018, 19eqeltrd 2839 . . . . . 6 ((𝜑𝑥𝐽) → (𝐹 “ (𝐹𝑥)) ∈ 𝐽)
21 dffn4 6283 . . . . . . . . 9 (𝐹 Fn 𝑋𝐹:𝑋onto→ran 𝐹)
224, 21sylib 208 . . . . . . . 8 (𝜑𝐹:𝑋onto→ran 𝐹)
23 elqtop3 21728 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto→ran 𝐹) → ((𝐹𝑥) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹𝑥) ⊆ ran 𝐹 ∧ (𝐹 “ (𝐹𝑥)) ∈ 𝐽)))
241, 22, 23syl2anc 696 . . . . . . 7 (𝜑 → ((𝐹𝑥) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹𝑥) ⊆ ran 𝐹 ∧ (𝐹 “ (𝐹𝑥)) ∈ 𝐽)))
2524adantr 472 . . . . . 6 ((𝜑𝑥𝐽) → ((𝐹𝑥) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹𝑥) ⊆ ran 𝐹 ∧ (𝐹 “ (𝐹𝑥)) ∈ 𝐽)))
2613, 20, 25mpbir2and 995 . . . . 5 ((𝜑𝑥𝐽) → (𝐹𝑥) ∈ (𝐽 qTop 𝐹))
2711, 26syl5eqel 2843 . . . 4 ((𝜑𝑥𝐽) → (𝐹𝑥) ∈ (𝐽 qTop 𝐹))
2827ralrimiva 3104 . . 3 (𝜑 → ∀𝑥𝐽 (𝐹𝑥) ∈ (𝐽 qTop 𝐹))
29 qtoptopon 21729 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto→ran 𝐹) → (𝐽 qTop 𝐹) ∈ (TopOn‘ran 𝐹))
301, 22, 29syl2anc 696 . . . 4 (𝜑 → (𝐽 qTop 𝐹) ∈ (TopOn‘ran 𝐹))
31 iscn 21261 . . . 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 696 . . 3 (𝜑 → (𝐹 ∈ ((𝐽 qTop 𝐹) Cn 𝐽) ↔ (𝐹:ran 𝐹𝑋 ∧ ∀𝑥𝐽 (𝐹𝑥) ∈ (𝐽 qTop 𝐹))))
3310, 28, 32mpbir2and 995 . 2 (𝜑𝐹 ∈ ((𝐽 qTop 𝐹) Cn 𝐽))
34 ishmeo 21784 . 2 (𝐹 ∈ (𝐽Homeo(𝐽 qTop 𝐹)) ↔ (𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)) ∧ 𝐹 ∈ ((𝐽 qTop 𝐹) Cn 𝐽)))
356, 33, 34sylanbrc 701 1 (𝜑𝐹 ∈ (𝐽Homeo(𝐽 qTop 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  wss 3715  ccnv 5265  ran crn 5267  cima 5269   Fn wfn 6044  wf 6045  1-1wf1 6046  ontowfo 6047  1-1-ontowf1o 6048  cfv 6049  (class class class)co 6814   qTop cqtop 16385  TopOnctopon 20937   Cn ccn 21250  Homeochmeo 21778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-map 8027  df-qtop 16389  df-top 20921  df-topon 20938  df-cn 21253  df-hmeo 21780
This theorem is referenced by:  t0kq  21843
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