Theorem qtopf1 23679

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 6739	. . . 4 ⊢ (𝐹:𝑋–1-1→𝑌 → 𝐹 Fn 𝑋)
4	2, 3	syl 17	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝑋)
5		qtopid 23568	. . 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 6793	. . . 4 ⊢ (𝐹:𝑋–1-1→𝑌 → 𝐹:𝑋–1-1-onto→ran 𝐹)
8		f1ocnv 6794	. . . 4 ⊢ (𝐹:𝑋–1-1-onto→ran 𝐹 → ◡𝐹:ran 𝐹–1-1-onto→𝑋)
9		f1of 6782	. . . 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 6167	. . . . 5 ⊢ (◡◡𝐹 “ 𝑥) = (𝐹 “ 𝑥)
12		imassrn 6031	. . . . . . 7 ⊢ (𝐹 “ 𝑥) ⊆ ran 𝐹
13	12	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ⊆ ran 𝐹)
14	2	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝐹:𝑋–1-1→𝑌)
15		toponss 22790	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋)
16	1, 15	sylan 580	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋)
17		f1imacnv 6798	. . . . . . . 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 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝑥 ∈ 𝐽)
20	18, 19	eqeltrd 2828	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (◡𝐹 “ (𝐹 “ 𝑥)) ∈ 𝐽)
21		dffn4 6760	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–onto→ran 𝐹)
22	4, 21	sylib 218	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑋–onto→ran 𝐹)
23		elqtop3 23566	. . . . . . . 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 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ((𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹 “ 𝑥) ⊆ ran 𝐹 ∧ (◡𝐹 “ (𝐹 “ 𝑥)) ∈ 𝐽)))
26	13, 20, 25	mpbir2and 713	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹))
27	11, 26	eqeltrid 2832	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (◡◡𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹))
28	27	ralrimiva 3125	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐽 (◡◡𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹))
29		qtoptopon 23567	. . . . 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 23098	. . . 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 713	. 2 ⊢ (𝜑 → ◡𝐹 ∈ ((𝐽 qTop 𝐹) Cn 𝐽))
34		ishmeo 23622	. 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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ⊆ wss 3911  ^◡ccnv 5630  ran crn 5632   “ cima 5634   Fn wfn 6494  ⟶wf 6495  –1-1→wf1 6496  –onto→wfo 6497  –1-1-onto→wf1o 6498  ‘cfv 6499  (class class class)co 7369   qTop cqtop 17442  TopOnctopon 22773   Cn ccn 23087  Homeochmeo 23616

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-map 8778  df-qtop 17446  df-top 22757  df-topon 22774  df-cn 23090  df-hmeo 23618

This theorem is referenced by:  t0kq  23681