Theorem qtopt1 33781

Description: If every equivalence class is closed, then the quotient space is T₁ . (Contributed by Thierry Arnoux, 5-Jan-2020.)

Hypotheses

Ref	Expression
qtopt1.x	⊢ 𝑋 = ∪ 𝐽
qtopt1.1	⊢ (𝜑 → 𝐽 ∈ Fre)
qtopt1.2	⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)
qtopt1.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (◡𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))

Assertion

Ref	Expression
qtopt1	⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ Fre)

Distinct variable groups:   𝑥,𝐹   𝑥,𝐽   𝜑,𝑥

Allowed substitution hints:   𝑋(𝑥)   𝑌(𝑥)

Proof of Theorem qtopt1

Step	Hyp	Ref	Expression
1		qtopt1.1	. . . 4 ⊢ (𝜑 → 𝐽 ∈ Fre)
2		t1top 23359	. . . 4 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Top)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → 𝐽 ∈ Top)
4		qtopt1.2	. . . 4 ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)
5		fofn 6836	. . . 4 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹 Fn 𝑋)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝑋)
7		qtopt1.x	. . . 4 ⊢ 𝑋 = ∪ 𝐽
8	7	qtoptop 23729	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Top)
9	3, 6, 8	syl2anc 583	. 2 ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ Top)
10		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → 𝑥 ∈ ∪ (𝐽 qTop 𝐹))
11	7	qtopuni 23731	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐹:𝑋–onto→𝑌) → 𝑌 = ∪ (𝐽 qTop 𝐹))
12	3, 4, 11	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → 𝑌 = ∪ (𝐽 qTop 𝐹))
13	12	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → 𝑌 = ∪ (𝐽 qTop 𝐹))
14	10, 13	eleqtrrd 2847	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → 𝑥 ∈ 𝑌)
15	14	snssd 4834	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → {𝑥} ⊆ 𝑌)
16		qtopt1.3	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (◡𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))
17	14, 16	syldan 590	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → (◡𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))
18	3, 7	jctir 520	. . . . . . 7 ⊢ (𝜑 → (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽))
19		istopon 22939	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽))
20	18, 19	sylibr 234	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
21		qtopcld 23742	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → ({𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ({𝑥} ⊆ 𝑌 ∧ (◡𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))))
22	20, 4, 21	syl2anc 583	. . . . 5 ⊢ (𝜑 → ({𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ({𝑥} ⊆ 𝑌 ∧ (◡𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))))
23	22	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → ({𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ({𝑥} ⊆ 𝑌 ∧ (◡𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))))
24	15, 17, 23	mpbir2and 712	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → {𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)))
25	24	ralrimiva 3152	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ∪ (𝐽 qTop 𝐹){𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)))
26		eqid 2740	. . 3 ⊢ ∪ (𝐽 qTop 𝐹) = ∪ (𝐽 qTop 𝐹)
27	26	ist1 23350	. 2 ⊢ ((𝐽 qTop 𝐹) ∈ Fre ↔ ((𝐽 qTop 𝐹) ∈ Top ∧ ∀𝑥 ∈ ∪ (𝐽 qTop 𝐹){𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹))))
28	9, 25, 27	sylanbrc 582	1 ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ Fre)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ⊆ wss 3976  {csn 4648  ∪ cuni 4931  ^◡ccnv 5699   “ cima 5703   Fn wfn 6568  –onto→wfo 6571  ‘cfv 6573  (class class class)co 7448   qTop cqtop 17563  Topctop 22920  TopOnctopon 22937  Clsdccld 23045  Frect1 23336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-qtop 17567  df-top 22921  df-topon 22938  df-cld 23048  df-t1 23343

This theorem is referenced by: (None)