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Theorem qtopf1 21529
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 6059 . . . 4 (𝐹:𝑋1-1𝑌𝐹 Fn 𝑋)
42, 3syl 17 . . 3 (𝜑𝐹 Fn 𝑋)
5 qtopid 21418 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
61, 4, 5syl2anc 692 . 2 (𝜑𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
7 f1f1orn 6105 . . . 4 (𝐹:𝑋1-1𝑌𝐹:𝑋1-1-onto→ran 𝐹)
8 f1ocnv 6106 . . . 4 (𝐹:𝑋1-1-onto→ran 𝐹𝐹:ran 𝐹1-1-onto𝑋)
9 f1of 6094 . . . 4 (𝐹:ran 𝐹1-1-onto𝑋𝐹:ran 𝐹𝑋)
102, 7, 8, 94syl 19 . . 3 (𝜑𝐹:ran 𝐹𝑋)
11 imacnvcnv 5558 . . . . 5 (𝐹𝑥) = (𝐹𝑥)
12 imassrn 5436 . . . . . . 7 (𝐹𝑥) ⊆ ran 𝐹
1312a1i 11 . . . . . 6 ((𝜑𝑥𝐽) → (𝐹𝑥) ⊆ ran 𝐹)
142adantr 481 . . . . . . . 8 ((𝜑𝑥𝐽) → 𝐹:𝑋1-1𝑌)
15 toponss 20644 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝐽) → 𝑥𝑋)
161, 15sylan 488 . . . . . . . 8 ((𝜑𝑥𝐽) → 𝑥𝑋)
17 f1imacnv 6110 . . . . . . . 8 ((𝐹:𝑋1-1𝑌𝑥𝑋) → (𝐹 “ (𝐹𝑥)) = 𝑥)
1814, 16, 17syl2anc 692 . . . . . . 7 ((𝜑𝑥𝐽) → (𝐹 “ (𝐹𝑥)) = 𝑥)
19 simpr 477 . . . . . . 7 ((𝜑𝑥𝐽) → 𝑥𝐽)
2018, 19eqeltrd 2698 . . . . . 6 ((𝜑𝑥𝐽) → (𝐹 “ (𝐹𝑥)) ∈ 𝐽)
21 dffn4 6078 . . . . . . . . 9 (𝐹 Fn 𝑋𝐹:𝑋onto→ran 𝐹)
224, 21sylib 208 . . . . . . . 8 (𝜑𝐹:𝑋onto→ran 𝐹)
23 elqtop3 21416 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto→ran 𝐹) → ((𝐹𝑥) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹𝑥) ⊆ ran 𝐹 ∧ (𝐹 “ (𝐹𝑥)) ∈ 𝐽)))
241, 22, 23syl2anc 692 . . . . . . 7 (𝜑 → ((𝐹𝑥) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹𝑥) ⊆ ran 𝐹 ∧ (𝐹 “ (𝐹𝑥)) ∈ 𝐽)))
2524adantr 481 . . . . . 6 ((𝜑𝑥𝐽) → ((𝐹𝑥) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹𝑥) ⊆ ran 𝐹 ∧ (𝐹 “ (𝐹𝑥)) ∈ 𝐽)))
2613, 20, 25mpbir2and 956 . . . . 5 ((𝜑𝑥𝐽) → (𝐹𝑥) ∈ (𝐽 qTop 𝐹))
2711, 26syl5eqel 2702 . . . 4 ((𝜑𝑥𝐽) → (𝐹𝑥) ∈ (𝐽 qTop 𝐹))
2827ralrimiva 2960 . . 3 (𝜑 → ∀𝑥𝐽 (𝐹𝑥) ∈ (𝐽 qTop 𝐹))
29 qtoptopon 21417 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto→ran 𝐹) → (𝐽 qTop 𝐹) ∈ (TopOn‘ran 𝐹))
301, 22, 29syl2anc 692 . . . 4 (𝜑 → (𝐽 qTop 𝐹) ∈ (TopOn‘ran 𝐹))
31 iscn 20949 . . . 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 692 . . 3 (𝜑 → (𝐹 ∈ ((𝐽 qTop 𝐹) Cn 𝐽) ↔ (𝐹:ran 𝐹𝑋 ∧ ∀𝑥𝐽 (𝐹𝑥) ∈ (𝐽 qTop 𝐹))))
3310, 28, 32mpbir2and 956 . 2 (𝜑𝐹 ∈ ((𝐽 qTop 𝐹) Cn 𝐽))
34 ishmeo 21472 . 2 (𝐹 ∈ (𝐽Homeo(𝐽 qTop 𝐹)) ↔ (𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)) ∧ 𝐹 ∈ ((𝐽 qTop 𝐹) Cn 𝐽)))
356, 33, 34sylanbrc 697 1 (𝜑𝐹 ∈ (𝐽Homeo(𝐽 qTop 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wss 3555  ccnv 5073  ran crn 5075  cima 5077   Fn wfn 5842  wf 5843  1-1wf1 5844  ontowfo 5845  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604   qTop cqtop 16084  TopOnctopon 20618   Cn ccn 20938  Homeochmeo 21466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-map 7804  df-qtop 16088  df-top 20621  df-topon 20623  df-cn 20941  df-hmeo 21468
This theorem is referenced by:  t0kq  21531
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