Theorem qtopt1 31687

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 22389	. . . 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 6674	. . . 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 22759	. . 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 22761	. . . . . . . 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 2842	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → 𝑥 ∈ 𝑌)
15	14	snssd 4739	. . . 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 21969	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽))
20	18, 19	sylibr 233	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
21		qtopcld 22772	. . . . . 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 709	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → {𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)))
25	24	ralrimiva 3107	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ∪ (𝐽 qTop 𝐹){𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)))
26		eqid 2738	. . 3 ⊢ ∪ (𝐽 qTop 𝐹) = ∪ (𝐽 qTop 𝐹)
27	26	ist1 22380	. 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 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063   ⊆ wss 3883  {csn 4558  ∪ cuni 4836  ^◡ccnv 5579   “ cima 5583   Fn wfn 6413  –onto→wfo 6416  ‘cfv 6418  (class class class)co 7255   qTop cqtop 17131  Topctop 21950  TopOnctopon 21967  Clsdccld 22075  Frect1 22366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-qtop 17135  df-top 21951  df-topon 21968  df-cld 22078  df-t1 22373

This theorem is referenced by: (None)