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Theorem qtopf1 21989
 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 6338 . . . 4 (𝐹:𝑋1-1𝑌𝐹 Fn 𝑋)
42, 3syl 17 . . 3 (𝜑𝐹 Fn 𝑋)
5 qtopid 21878 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
61, 4, 5syl2anc 581 . 2 (𝜑𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
7 f1f1orn 6388 . . . 4 (𝐹:𝑋1-1𝑌𝐹:𝑋1-1-onto→ran 𝐹)
8 f1ocnv 6389 . . . 4 (𝐹:𝑋1-1-onto→ran 𝐹𝐹:ran 𝐹1-1-onto𝑋)
9 f1of 6377 . . . 4 (𝐹:ran 𝐹1-1-onto𝑋𝐹:ran 𝐹𝑋)
102, 7, 8, 94syl 19 . . 3 (𝜑𝐹:ran 𝐹𝑋)
11 imacnvcnv 5839 . . . . 5 (𝐹𝑥) = (𝐹𝑥)
12 imassrn 5717 . . . . . . 7 (𝐹𝑥) ⊆ ran 𝐹
1312a1i 11 . . . . . 6 ((𝜑𝑥𝐽) → (𝐹𝑥) ⊆ ran 𝐹)
142adantr 474 . . . . . . . 8 ((𝜑𝑥𝐽) → 𝐹:𝑋1-1𝑌)
15 toponss 21101 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝐽) → 𝑥𝑋)
161, 15sylan 577 . . . . . . . 8 ((𝜑𝑥𝐽) → 𝑥𝑋)
17 f1imacnv 6393 . . . . . . . 8 ((𝐹:𝑋1-1𝑌𝑥𝑋) → (𝐹 “ (𝐹𝑥)) = 𝑥)
1814, 16, 17syl2anc 581 . . . . . . 7 ((𝜑𝑥𝐽) → (𝐹 “ (𝐹𝑥)) = 𝑥)
19 simpr 479 . . . . . . 7 ((𝜑𝑥𝐽) → 𝑥𝐽)
2018, 19eqeltrd 2905 . . . . . 6 ((𝜑𝑥𝐽) → (𝐹 “ (𝐹𝑥)) ∈ 𝐽)
21 dffn4 6358 . . . . . . . . 9 (𝐹 Fn 𝑋𝐹:𝑋onto→ran 𝐹)
224, 21sylib 210 . . . . . . . 8 (𝜑𝐹:𝑋onto→ran 𝐹)
23 elqtop3 21876 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto→ran 𝐹) → ((𝐹𝑥) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹𝑥) ⊆ ran 𝐹 ∧ (𝐹 “ (𝐹𝑥)) ∈ 𝐽)))
241, 22, 23syl2anc 581 . . . . . . 7 (𝜑 → ((𝐹𝑥) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹𝑥) ⊆ ran 𝐹 ∧ (𝐹 “ (𝐹𝑥)) ∈ 𝐽)))
2524adantr 474 . . . . . 6 ((𝜑𝑥𝐽) → ((𝐹𝑥) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹𝑥) ⊆ ran 𝐹 ∧ (𝐹 “ (𝐹𝑥)) ∈ 𝐽)))
2613, 20, 25mpbir2and 706 . . . . 5 ((𝜑𝑥𝐽) → (𝐹𝑥) ∈ (𝐽 qTop 𝐹))
2711, 26syl5eqel 2909 . . . 4 ((𝜑𝑥𝐽) → (𝐹𝑥) ∈ (𝐽 qTop 𝐹))
2827ralrimiva 3174 . . 3 (𝜑 → ∀𝑥𝐽 (𝐹𝑥) ∈ (𝐽 qTop 𝐹))
29 qtoptopon 21877 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto→ran 𝐹) → (𝐽 qTop 𝐹) ∈ (TopOn‘ran 𝐹))
301, 22, 29syl2anc 581 . . . 4 (𝜑 → (𝐽 qTop 𝐹) ∈ (TopOn‘ran 𝐹))
31 iscn 21409 . . . 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 581 . . 3 (𝜑 → (𝐹 ∈ ((𝐽 qTop 𝐹) Cn 𝐽) ↔ (𝐹:ran 𝐹𝑋 ∧ ∀𝑥𝐽 (𝐹𝑥) ∈ (𝐽 qTop 𝐹))))
3310, 28, 32mpbir2and 706 . 2 (𝜑𝐹 ∈ ((𝐽 qTop 𝐹) Cn 𝐽))
34 ishmeo 21932 . 2 (𝐹 ∈ (𝐽Homeo(𝐽 qTop 𝐹)) ↔ (𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)) ∧ 𝐹 ∈ ((𝐽 qTop 𝐹) Cn 𝐽)))
356, 33, 34sylanbrc 580 1 (𝜑𝐹 ∈ (𝐽Homeo(𝐽 qTop 𝐹)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1658   ∈ wcel 2166  ∀wral 3116   ⊆ wss 3797  ◡ccnv 5340  ran crn 5342   “ cima 5344   Fn wfn 6117  ⟶wf 6118  –1-1→wf1 6119  –onto→wfo 6120  –1-1-onto→wf1o 6121  ‘cfv 6122  (class class class)co 6904   qTop cqtop 16515  TopOnctopon 21084   Cn ccn 21398  Homeochmeo 21926 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-rep 4993  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-ral 3121  df-rex 3122  df-reu 3123  df-rab 3125  df-v 3415  df-sbc 3662  df-csb 3757  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-iun 4741  df-br 4873  df-opab 4935  df-mpt 4952  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-f1 6127  df-fo 6128  df-f1o 6129  df-fv 6130  df-ov 6907  df-oprab 6908  df-mpt2 6909  df-map 8123  df-qtop 16519  df-top 21068  df-topon 21085  df-cn 21401  df-hmeo 21928 This theorem is referenced by:  t0kq  21991
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