Theorem qtopf1 23941

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 6776	. . . 4 ⊢ (𝐹:𝑋–1-1→𝑌 → 𝐹 Fn 𝑋)
4	2, 3	syl 18	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝑋)
5		qtopid 23830	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
6	1, 4, 5	syl2anc 595	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
7		f1f1orn 6833	. . . 4 ⊢ (𝐹:𝑋–1-1→𝑌 → 𝐹:𝑋–1-1-onto→ran 𝐹)
8		f1ocnv 6834	. . . 4 ⊢ (𝐹:𝑋–1-1-onto→ran 𝐹 → ◡𝐹:ran 𝐹–1-1-onto→𝑋)
9		f1of 6821	. . . 4 ⊢ (◡𝐹:ran 𝐹–1-1-onto→𝑋 → ◡𝐹:ran 𝐹⟶𝑋)
10	2, 7, 8, 9	4syl 20	. . 3 ⊢ (𝜑 → ◡𝐹:ran 𝐹⟶𝑋)
11		imacnvcnv 6208	. . . . 5 ⊢ (◡◡𝐹 “ 𝑥) = (𝐹 “ 𝑥)
12		imassrn 6074	. . . . . . 7 ⊢ (𝐹 “ 𝑥) ⊆ ran 𝐹
13	12	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ⊆ ran 𝐹)
14	2	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝐹:𝑋–1-1→𝑌)
15		toponss 23052	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋)
16	1, 15	sylan 591	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋)
17		f1imacnv 6838	. . . . . . . 8 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ 𝑥 ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ 𝑥)) = 𝑥)
18	14, 16, 17	syl2anc 595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (◡𝐹 “ (𝐹 “ 𝑥)) = 𝑥)
19		simpr 489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝑥 ∈ 𝐽)
20	18, 19	eqeltrd 2869	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (◡𝐹 “ (𝐹 “ 𝑥)) ∈ 𝐽)
21		dffn4 6799	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–onto→ran 𝐹)
22	4, 21	sylib 221	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑋–onto→ran 𝐹)
23		elqtop3 23828	. . . . . . . 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 595	. . . . . . 7 ⊢ (𝜑 → ((𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹 “ 𝑥) ⊆ ran 𝐹 ∧ (◡𝐹 “ (𝐹 “ 𝑥)) ∈ 𝐽)))
25	24	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ((𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹 “ 𝑥) ⊆ ran 𝐹 ∧ (◡𝐹 “ (𝐹 “ 𝑥)) ∈ 𝐽)))
26	13, 20, 25	mpbir2and 725	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹))
27	11, 26	eqeltrid 2873	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (◡◡𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹))
28	27	ralrimiva 3163	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐽 (◡◡𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹))
29		qtoptopon 23829	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→ran 𝐹) → (𝐽 qTop 𝐹) ∈ (TopOn‘ran 𝐹))
30	1, 22, 29	syl2anc 595	. . . 4 ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ (TopOn‘ran 𝐹))
31		iscn 23360	. . . 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 595	. . 3 ⊢ (𝜑 → (◡𝐹 ∈ ((𝐽 qTop 𝐹) Cn 𝐽) ↔ (◡𝐹:ran 𝐹⟶𝑋 ∧ ∀𝑥 ∈ 𝐽 (◡◡𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹))))
33	10, 28, 32	mpbir2and 725	. 2 ⊢ (𝜑 → ◡𝐹 ∈ ((𝐽 qTop 𝐹) Cn 𝐽))
34		ishmeo 23884	. 2 ⊢ (𝐹 ∈ (𝐽Homeo(𝐽 qTop 𝐹)) ↔ (𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)) ∧ ◡𝐹 ∈ ((𝐽 qTop 𝐹) Cn 𝐽)))
35	6, 33, 34	sylanbrc 594	1 ⊢ (𝜑 → 𝐹 ∈ (𝐽Homeo(𝐽 qTop 𝐹)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085   ⊆ wss 3913  ^◡ccnv 5661  ran crn 5663   “ cima 5665   Fn wfn 6532  ⟶wf 6533  –1-1→wf1 6534  –onto→wfo 6535  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7411   qTop cqtop 17556  TopOnctopon 23035   Cn ccn 23349  Homeochmeo 23878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-map 8825  df-qtop 17560  df-top 23019  df-topon 23036  df-cn 23352  df-hmeo 23880

This theorem is referenced by:  t0kq  23943