Theorem qtopt1 33866

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 23268	. . . 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 6792	. . . 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 23638	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Top)
9	3, 6, 8	syl2anc 584	. 2 ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ Top)
10		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → 𝑥 ∈ ∪ (𝐽 qTop 𝐹))
11	7	qtopuni 23640	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐹:𝑋–onto→𝑌) → 𝑌 = ∪ (𝐽 qTop 𝐹))
12	3, 4, 11	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → 𝑌 = ∪ (𝐽 qTop 𝐹))
13	12	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → 𝑌 = ∪ (𝐽 qTop 𝐹))
14	10, 13	eleqtrrd 2837	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → 𝑥 ∈ 𝑌)
15	14	snssd 4785	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → {𝑥} ⊆ 𝑌)
16		qtopt1.3	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (◡𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))
17	14, 16	syldan 591	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → (◡𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))
18	3, 7	jctir 520	. . . . . . 7 ⊢ (𝜑 → (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽))
19		istopon 22850	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽))
20	18, 19	sylibr 234	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
21		qtopcld 23651	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → ({𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ({𝑥} ⊆ 𝑌 ∧ (◡𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))))
22	20, 4, 21	syl2anc 584	. . . . 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 713	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → {𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)))
25	24	ralrimiva 3132	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ∪ (𝐽 qTop 𝐹){𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)))
26		eqid 2735	. . 3 ⊢ ∪ (𝐽 qTop 𝐹) = ∪ (𝐽 qTop 𝐹)
27	26	ist1 23259	. 2 ⊢ ((𝐽 qTop 𝐹) ∈ Fre ↔ ((𝐽 qTop 𝐹) ∈ Top ∧ ∀𝑥 ∈ ∪ (𝐽 qTop 𝐹){𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹))))
28	9, 25, 27	sylanbrc 583	1 ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ Fre)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3051   ⊆ wss 3926  {csn 4601  ∪ cuni 4883  ^◡ccnv 5653   “ cima 5657   Fn wfn 6526  –onto→wfo 6529  ‘cfv 6531  (class class class)co 7405   qTop cqtop 17517  Topctop 22831  TopOnctopon 22848  Clsdccld 22954  Frect1 23245

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-qtop 17521  df-top 22832  df-topon 22849  df-cld 22957  df-t1 23252

This theorem is referenced by: (None)