Theorem qtopf1 22967

Description: If a quotient map is injective, then it is a homeomorphism. (Contributed by Mario Carneiro, 25-Aug-2015.)

Hypotheses

Ref	Expression
qtopf1.1	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
qtopf1.2	⊢ (𝜑 → 𝐹:𝑋–1-1→𝑌)

Assertion

Ref	Expression
qtopf1	⊢ (𝜑 → 𝐹 ∈ (𝐽Homeo(𝐽 qTop 𝐹)))

Proof of Theorem qtopf1

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		qtopf1.1	. . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
2		qtopf1.2	. . . 4 ⊢ (𝜑 → 𝐹:𝑋–1-1→𝑌)
3		f1fn 6671	. . . 4 ⊢ (𝐹:𝑋–1-1→𝑌 → 𝐹 Fn 𝑋)
4	2, 3	syl 17	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝑋)
5		qtopid 22856	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
6	1, 4, 5	syl2anc 584	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
7		f1f1orn 6727	. . . 4 ⊢ (𝐹:𝑋–1-1→𝑌 → 𝐹:𝑋–1-1-onto→ran 𝐹)
8		f1ocnv 6728	. . . 4 ⊢ (𝐹:𝑋–1-1-onto→ran 𝐹 → ◡𝐹:ran 𝐹–1-1-onto→𝑋)
9		f1of 6716	. . . 4 ⊢ (◡𝐹:ran 𝐹–1-1-onto→𝑋 → ◡𝐹:ran 𝐹⟶𝑋)
10	2, 7, 8, 9	4syl 19	. . 3 ⊢ (𝜑 → ◡𝐹:ran 𝐹⟶𝑋)
11		imacnvcnv 6109	. . . . 5 ⊢ (◡◡𝐹 “ 𝑥) = (𝐹 “ 𝑥)
12		imassrn 5980	. . . . . . 7 ⊢ (𝐹 “ 𝑥) ⊆ ran 𝐹
13	12	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ⊆ ran 𝐹)
14	2	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝐹:𝑋–1-1→𝑌)
15		toponss 22076	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋)
16	1, 15	sylan 580	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋)
17		f1imacnv 6732	. . . . . . . 8 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ 𝑥 ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ 𝑥)) = 𝑥)
18	14, 16, 17	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (◡𝐹 “ (𝐹 “ 𝑥)) = 𝑥)
19		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝑥 ∈ 𝐽)
20	18, 19	eqeltrd 2839	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (◡𝐹 “ (𝐹 “ 𝑥)) ∈ 𝐽)
21		dffn4 6694	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–onto→ran 𝐹)
22	4, 21	sylib 217	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑋–onto→ran 𝐹)
23		elqtop3 22854	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→ran 𝐹) → ((𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹 “ 𝑥) ⊆ ran 𝐹 ∧ (◡𝐹 “ (𝐹 “ 𝑥)) ∈ 𝐽)))
24	1, 22, 23	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ((𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹 “ 𝑥) ⊆ ran 𝐹 ∧ (◡𝐹 “ (𝐹 “ 𝑥)) ∈ 𝐽)))
25	24	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ((𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹 “ 𝑥) ⊆ ran 𝐹 ∧ (◡𝐹 “ (𝐹 “ 𝑥)) ∈ 𝐽)))
26	13, 20, 25	mpbir2and 710	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹))
27	11, 26	eqeltrid 2843	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (◡◡𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹))
28	27	ralrimiva 3103	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐽 (◡◡𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹))
29		qtoptopon 22855	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→ran 𝐹) → (𝐽 qTop 𝐹) ∈ (TopOn‘ran 𝐹))
30	1, 22, 29	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ (TopOn‘ran 𝐹))
31		iscn 22386	. . . 4 ⊢ (((𝐽 qTop 𝐹) ∈ (TopOn‘ran 𝐹) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (◡𝐹 ∈ ((𝐽 qTop 𝐹) Cn 𝐽) ↔ (◡𝐹:ran 𝐹⟶𝑋 ∧ ∀𝑥 ∈ 𝐽 (◡◡𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹))))
32	30, 1, 31	syl2anc 584	. . 3 ⊢ (𝜑 → (◡𝐹 ∈ ((𝐽 qTop 𝐹) Cn 𝐽) ↔ (◡𝐹:ran 𝐹⟶𝑋 ∧ ∀𝑥 ∈ 𝐽 (◡◡𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹))))
33	10, 28, 32	mpbir2and 710	. 2 ⊢ (𝜑 → ◡𝐹 ∈ ((𝐽 qTop 𝐹) Cn 𝐽))
34		ishmeo 22910	. 2 ⊢ (𝐹 ∈ (𝐽Homeo(𝐽 qTop 𝐹)) ↔ (𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)) ∧ ◡𝐹 ∈ ((𝐽 qTop 𝐹) Cn 𝐽)))
35	6, 33, 34	sylanbrc 583	1 ⊢ (𝜑 → 𝐹 ∈ (𝐽Homeo(𝐽 qTop 𝐹)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064   ⊆ wss 3887  ^◡ccnv 5588  ran crn 5590   “ cima 5592   Fn wfn 6428  ⟶wf 6429  –1-1→wf1 6430  –onto→wfo 6431  –1-1-onto→wf1o 6432  ‘cfv 6433  (class class class)co 7275   qTop cqtop 17214  TopOnctopon 22059   Cn ccn 22375  Homeochmeo 22904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617  df-qtop 17218  df-top 22043  df-topon 22060  df-cn 22378  df-hmeo 22906

This theorem is referenced by:  t0kq  22969