Theorem qtopf1 23772

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 23661	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
6	1, 4, 5	syl2anc 585	. 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 6172	. . . . 5 ⊢ (◡◡𝐹 “ 𝑥) = (𝐹 “ 𝑥)
12		imassrn 6038	. . . . . . 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 22883	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋)
16	1, 15	sylan 581	. . . . . . . 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 585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (◡𝐹 “ (𝐹 “ 𝑥)) = 𝑥)
19		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝑥 ∈ 𝐽)
20	18, 19	eqeltrd 2837	. . . . . 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 23659	. . . . . . . 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 585	. . . . . . 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 714	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹))
27	11, 26	eqeltrid 2841	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (◡◡𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹))
28	27	ralrimiva 3130	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐽 (◡◡𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹))
29		qtoptopon 23660	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→ran 𝐹) → (𝐽 qTop 𝐹) ∈ (TopOn‘ran 𝐹))
30	1, 22, 29	syl2anc 585	. . . 4 ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ (TopOn‘ran 𝐹))
31		iscn 23191	. . . 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 585	. . 3 ⊢ (𝜑 → (◡𝐹 ∈ ((𝐽 qTop 𝐹) Cn 𝐽) ↔ (◡𝐹:ran 𝐹⟶𝑋 ∧ ∀𝑥 ∈ 𝐽 (◡◡𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹))))
33	10, 28, 32	mpbir2and 714	. 2 ⊢ (𝜑 → ◡𝐹 ∈ ((𝐽 qTop 𝐹) Cn 𝐽))
34		ishmeo 23715	. 2 ⊢ (𝐹 ∈ (𝐽Homeo(𝐽 qTop 𝐹)) ↔ (𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)) ∧ ◡𝐹 ∈ ((𝐽 qTop 𝐹) Cn 𝐽)))
35	6, 33, 34	sylanbrc 584	1 ⊢ (𝜑 → 𝐹 ∈ (𝐽Homeo(𝐽 qTop 𝐹)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3903  ^◡ccnv 5631  ran crn 5633   “ cima 5635   Fn wfn 6495  ⟶wf 6496  –1-1→wf1 6497  –onto→wfo 6498  –1-1-onto→wf1o 6499  ‘cfv 6500  (class class class)co 7368   qTop cqtop 17436  TopOnctopon 22866   Cn ccn 23180  Homeochmeo 23709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-map 8777  df-qtop 17440  df-top 22850  df-topon 22867  df-cn 23183  df-hmeo 23711

This theorem is referenced by:  t0kq  23774