Theorem qtopf1 21619

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 6102	. . . 4 ⊢ (𝐹:𝑋–1-1→𝑌 → 𝐹 Fn 𝑋)
4	2, 3	syl 17	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝑋)
5		qtopid 21508	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
6	1, 4, 5	syl2anc 693	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
7		f1f1orn 6148	. . . 4 ⊢ (𝐹:𝑋–1-1→𝑌 → 𝐹:𝑋–1-1-onto→ran 𝐹)
8		f1ocnv 6149	. . . 4 ⊢ (𝐹:𝑋–1-1-onto→ran 𝐹 → ◡𝐹:ran 𝐹–1-1-onto→𝑋)
9		f1of 6137	. . . 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 5599	. . . . 5 ⊢ (◡◡𝐹 “ 𝑥) = (𝐹 “ 𝑥)
12		imassrn 5477	. . . . . . 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 20731	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋)
16	1, 15	sylan 488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋)
17		f1imacnv 6153	. . . . . . . 8 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ 𝑥 ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ 𝑥)) = 𝑥)
18	14, 16, 17	syl2anc 693	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (◡𝐹 “ (𝐹 “ 𝑥)) = 𝑥)
19		simpr 477	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝑥 ∈ 𝐽)
20	18, 19	eqeltrd 2701	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (◡𝐹 “ (𝐹 “ 𝑥)) ∈ 𝐽)
21		dffn4 6121	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–onto→ran 𝐹)
22	4, 21	sylib 208	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑋–onto→ran 𝐹)
23		elqtop3 21506	. . . . . . . 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 693	. . . . . . 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 957	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹))
27	11, 26	syl5eqel 2705	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (◡◡𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹))
28	27	ralrimiva 2966	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐽 (◡◡𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹))
29		qtoptopon 21507	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→ran 𝐹) → (𝐽 qTop 𝐹) ∈ (TopOn‘ran 𝐹))
30	1, 22, 29	syl2anc 693	. . . 4 ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ (TopOn‘ran 𝐹))
31		iscn 21039	. . . 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 693	. . 3 ⊢ (𝜑 → (◡𝐹 ∈ ((𝐽 qTop 𝐹) Cn 𝐽) ↔ (◡𝐹:ran 𝐹⟶𝑋 ∧ ∀𝑥 ∈ 𝐽 (◡◡𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹))))
33	10, 28, 32	mpbir2and 957	. 2 ⊢ (𝜑 → ◡𝐹 ∈ ((𝐽 qTop 𝐹) Cn 𝐽))
34		ishmeo 21562	. 2 ⊢ (𝐹 ∈ (𝐽Homeo(𝐽 qTop 𝐹)) ↔ (𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)) ∧ ◡𝐹 ∈ ((𝐽 qTop 𝐹) Cn 𝐽)))
35	6, 33, 34	sylanbrc 698	1 ⊢ (𝜑 → 𝐹 ∈ (𝐽Homeo(𝐽 qTop 𝐹)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912   ⊆ wss 3574  ^◡ccnv 5113  ran crn 5115   “ cima 5117   Fn wfn 5883  ⟶wf 5884  –1-1→wf1 5885  –onto→wfo 5886  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   qTop cqtop 16163  TopOnctopon 20715   Cn ccn 21028  Homeochmeo 21556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-qtop 16167  df-top 20699  df-topon 20716  df-cn 21031  df-hmeo 21558

This theorem is referenced by:  t0kq  21621