Theorem qtopt1 34332

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 23556	. . . 4 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Top)
3	1, 2	syl 18	. . 3 ⊢ (𝜑 → 𝐽 ∈ Top)
4		qtopt1.2	. . . 4 ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)
5		fofn 6795	. . . 4 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹 Fn 𝑋)
6	4, 5	syl 18	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝑋)
7		qtopt1.x	. . . 4 ⊢ 𝑋 = ∪ 𝐽
8	7	qtoptop 23927	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Top)
9	3, 6, 8	syl2anc 596	. 2 ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ Top)
10		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → 𝑥 ∈ ∪ (𝐽 qTop 𝐹))
11	7	qtopuni 23929	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐹:𝑋–onto→𝑌) → 𝑌 = ∪ (𝐽 qTop 𝐹))
12	3, 4, 11	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → 𝑌 = ∪ (𝐽 qTop 𝐹))
13	12	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → 𝑌 = ∪ (𝐽 qTop 𝐹))
14	10, 13	eleqtrrd 2865	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → 𝑥 ∈ 𝑌)
15	14	snssd 4750	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → {𝑥} ⊆ 𝑌)
16		qtopt1.3	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (◡𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))
17	14, 16	syldan 603	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → (◡𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))
18	3, 7	jctir 530	. . . . . . 7 ⊢ (𝜑 → (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽))
19		istopon 23138	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽))
20	18, 19	sylibr 237	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
21		qtopcld 23940	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → ({𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ({𝑥} ⊆ 𝑌 ∧ (◡𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))))
22	20, 4, 21	syl2anc 596	. . . . 5 ⊢ (𝜑 → ({𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ({𝑥} ⊆ 𝑌 ∧ (◡𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))))
23	22	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → ({𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ({𝑥} ⊆ 𝑌 ∧ (◡𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))))
24	15, 17, 23	mpbir2and 726	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐽 qTop 𝐹)) → {𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)))
25	24	ralrimiva 3156	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ∪ (𝐽 qTop 𝐹){𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹)))
26		eqid 2762	. . 3 ⊢ ∪ (𝐽 qTop 𝐹) = ∪ (𝐽 qTop 𝐹)
27	26	ist1 23547	. 2 ⊢ ((𝐽 qTop 𝐹) ∈ Fre ↔ ((𝐽 qTop 𝐹) ∈ Top ∧ ∀𝑥 ∈ ∪ (𝐽 qTop 𝐹){𝑥} ∈ (Clsd‘(𝐽 qTop 𝐹))))
28	9, 25, 27	sylanbrc 595	1 ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ Fre)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078   ⊆ wss 3902  {csn 4587  ∪ cuni 4870  ^◡ccnv 5658   “ cima 5662   Fn wfn 6532  –onto→wfo 6535  ‘cfv 6537  (class class class)co 7416   qTop cqtop 17593  Topctop 23119  TopOnctopon 23136  Clsdccld 23242  Frect1 23533

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-qtop 17597  df-top 23120  df-topon 23137  df-cld 23245  df-t1 23540

This theorem is used by: (None)