Theorem qtophaus 29231

Description: If an open map's graph in the product space (𝐽 ×_t 𝐽) is closed, then its quotient topology is Hausdorff. (Contributed by Thierry Arnoux, 4-Jan-2020.)

Hypotheses

Ref	Expression
qtophaus.x	⊢ 𝑋 = ∪ 𝐽
qtophaus.e	⊢ ∼ = (◡𝐹 ∘ 𝐹)
qtophaus.h	⊢ 𝐻 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
qtophaus.1	⊢ (𝜑 → 𝐽 ∈ Haus)
qtophaus.2	⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)
qtophaus.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹))
qtophaus.4	⊢ (𝜑 → ∼ ∈ (Clsd‘(𝐽 ×t 𝐽)))

Assertion

Ref	Expression
qtophaus	⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ Haus)

Distinct variable groups:   𝑥, ∼ ,𝑦   𝑥,𝐹,𝑦   𝑥,𝐻,𝑦   𝑥,𝐽,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝜑,𝑥,𝑦

Proof of Theorem qtophaus

Dummy variables 𝑎 𝑏 𝑐 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		qtophaus.1	. . . 4 ⊢ (𝜑 → 𝐽 ∈ Haus)
2		haustop 20945	. . . 4 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → 𝐽 ∈ Top)
4		qtophaus.2	. . . 4 ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)
5		fofn 6030	. . . 4 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹 Fn 𝑋)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝑋)
7		qtophaus.x	. . . 4 ⊢ 𝑋 = ∪ 𝐽
8	7	qtoptop 21313	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Top)
9	3, 6, 8	syl2anc 691	. 2 ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ Top)
10		txtop 21182	. . . 4 ⊢ (((𝐽 qTop 𝐹) ∈ Top ∧ (𝐽 qTop 𝐹) ∈ Top) → ((𝐽 qTop 𝐹) ×t (𝐽 qTop 𝐹)) ∈ Top)
11	9, 9, 10	syl2anc 691	. . 3 ⊢ (𝜑 → ((𝐽 qTop 𝐹) ×t (𝐽 qTop 𝐹)) ∈ Top)
12		idssxp 28811	. . . 4 ⊢ ( I ↾ ∪ (𝐽 qTop 𝐹)) ⊆ (∪ (𝐽 qTop 𝐹) × ∪ (𝐽 qTop 𝐹))
13		eqid 2610	. . . . . 6 ⊢ ∪ (𝐽 qTop 𝐹) = ∪ (𝐽 qTop 𝐹)
14	13, 13	txuni 21205	. . . . 5 ⊢ (((𝐽 qTop 𝐹) ∈ Top ∧ (𝐽 qTop 𝐹) ∈ Top) → (∪ (𝐽 qTop 𝐹) × ∪ (𝐽 qTop 𝐹)) = ∪ ((𝐽 qTop 𝐹) ×t (𝐽 qTop 𝐹)))
15	9, 9, 14	syl2anc 691	. . . 4 ⊢ (𝜑 → (∪ (𝐽 qTop 𝐹) × ∪ (𝐽 qTop 𝐹)) = ∪ ((𝐽 qTop 𝐹) ×t (𝐽 qTop 𝐹)))
16	12, 15	syl5sseq 3616	. . 3 ⊢ (𝜑 → ( I ↾ ∪ (𝐽 qTop 𝐹)) ⊆ ∪ ((𝐽 qTop 𝐹) ×t (𝐽 qTop 𝐹)))
17	7	qtopuni 21315	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐹:𝑋–onto→𝑌) → 𝑌 = ∪ (𝐽 qTop 𝐹))
18	3, 4, 17	syl2anc 691	. . . . . . 7 ⊢ (𝜑 → 𝑌 = ∪ (𝐽 qTop 𝐹))
19	18	sqxpeqd 5065	. . . . . 6 ⊢ (𝜑 → (𝑌 × 𝑌) = (∪ (𝐽 qTop 𝐹) × ∪ (𝐽 qTop 𝐹)))
20	19, 15	eqtr2d 2645	. . . . 5 ⊢ (𝜑 → ∪ ((𝐽 qTop 𝐹) ×t (𝐽 qTop 𝐹)) = (𝑌 × 𝑌))
21	18	eqcomd 2616	. . . . . 6 ⊢ (𝜑 → ∪ (𝐽 qTop 𝐹) = 𝑌)
22	21	reseq2d 5317	. . . . 5 ⊢ (𝜑 → ( I ↾ ∪ (𝐽 qTop 𝐹)) = ( I ↾ 𝑌))
23	20, 22	difeq12d 3691	. . . 4 ⊢ (𝜑 → (∪ ((𝐽 qTop 𝐹) ×t (𝐽 qTop 𝐹)) ∖ ( I ↾ ∪ (𝐽 qTop 𝐹))) = ((𝑌 × 𝑌) ∖ ( I ↾ 𝑌)))
24		simp-4r 803	. . . . . . . . . . . . . . . 16 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) ∧ 𝑎 ∈ 𝑌) ∧ 𝑏 ∈ 𝑌) ∧ 𝑐 = ⟨𝑎, 𝑏⟩) ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = 𝑎) ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑦) = 𝑏) → 𝑥 ∈ 𝑋)
25		simplr 788	. . . . . . . . . . . . . . . 16 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) ∧ 𝑎 ∈ 𝑌) ∧ 𝑏 ∈ 𝑌) ∧ 𝑐 = ⟨𝑎, 𝑏⟩) ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = 𝑎) ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑦) = 𝑏) → 𝑦 ∈ 𝑋)
26		opelxpi 5072	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑋))
27	24, 25, 26	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) ∧ 𝑎 ∈ 𝑌) ∧ 𝑏 ∈ 𝑌) ∧ 𝑐 = ⟨𝑎, 𝑏⟩) ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = 𝑎) ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑦) = 𝑏) → ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑋))
28		df-br 4584	. . . . . . . . . . . . . . 15 ⊢ (𝑥(𝑋 × 𝑋)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑋))
29	27, 28	sylibr 223	. . . . . . . . . . . . . 14 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) ∧ 𝑎 ∈ 𝑌) ∧ 𝑏 ∈ 𝑌) ∧ 𝑐 = ⟨𝑎, 𝑏⟩) ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = 𝑎) ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑦) = 𝑏) → 𝑥(𝑋 × 𝑋)𝑦)
30		simpllr 795	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) ∧ 𝑎 ∈ 𝑌) ∧ 𝑏 ∈ 𝑌) ∧ 𝑐 = ⟨𝑎, 𝑏⟩) ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = 𝑎) ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑦) = 𝑏) → (𝐹‘𝑥) = 𝑎)
31		simpr 476	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) ∧ 𝑎 ∈ 𝑌) ∧ 𝑏 ∈ 𝑌) ∧ 𝑐 = ⟨𝑎, 𝑏⟩) ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = 𝑎) ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑦) = 𝑏) → (𝐹‘𝑦) = 𝑏)
32	30, 31	opeq12d 4348	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) ∧ 𝑎 ∈ 𝑌) ∧ 𝑏 ∈ 𝑌) ∧ 𝑐 = ⟨𝑎, 𝑏⟩) ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = 𝑎) ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑦) = 𝑏) → ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ = ⟨𝑎, 𝑏⟩)
33		simp-5r 805	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) ∧ 𝑎 ∈ 𝑌) ∧ 𝑏 ∈ 𝑌) ∧ 𝑐 = ⟨𝑎, 𝑏⟩) ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = 𝑎) ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑦) = 𝑏) → 𝑐 = ⟨𝑎, 𝑏⟩)
34		simp-8r 811	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) ∧ 𝑎 ∈ 𝑌) ∧ 𝑏 ∈ 𝑌) ∧ 𝑐 = ⟨𝑎, 𝑏⟩) ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = 𝑎) ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑦) = 𝑏) → 𝑐 ∈ ((𝑌 × 𝑌) ∖ I ))
35	33, 34	eqeltrrd 2689	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) ∧ 𝑎 ∈ 𝑌) ∧ 𝑏 ∈ 𝑌) ∧ 𝑐 = ⟨𝑎, 𝑏⟩) ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = 𝑎) ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑦) = 𝑏) → ⟨𝑎, 𝑏⟩ ∈ ((𝑌 × 𝑌) ∖ I ))
36	32, 35	eqeltrd 2688	. . . . . . . . . . . . . . . . 17 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) ∧ 𝑎 ∈ 𝑌) ∧ 𝑏 ∈ 𝑌) ∧ 𝑐 = ⟨𝑎, 𝑏⟩) ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = 𝑎) ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑦) = 𝑏) → ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ ∈ ((𝑌 × 𝑌) ∖ I ))
37		relxp 5150	. . . . . . . . . . . . . . . . . 18 ⊢ Rel (𝑌 × 𝑌)
38		opeldifid 28794	. . . . . . . . . . . . . . . . . 18 ⊢ (Rel (𝑌 × 𝑌) → (⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ ∈ ((𝑌 × 𝑌) ∖ I ) ↔ (⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ ∈ (𝑌 × 𝑌) ∧ (𝐹‘𝑥) ≠ (𝐹‘𝑦))))
39	37, 38	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ ∈ ((𝑌 × 𝑌) ∖ I ) ↔ (⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ ∈ (𝑌 × 𝑌) ∧ (𝐹‘𝑥) ≠ (𝐹‘𝑦)))
40	36, 39	sylib 207	. . . . . . . . . . . . . . . 16 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) ∧ 𝑎 ∈ 𝑌) ∧ 𝑏 ∈ 𝑌) ∧ 𝑐 = ⟨𝑎, 𝑏⟩) ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = 𝑎) ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑦) = 𝑏) → (⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ ∈ (𝑌 × 𝑌) ∧ (𝐹‘𝑥) ≠ (𝐹‘𝑦)))
41	40	simprd 478	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) ∧ 𝑎 ∈ 𝑌) ∧ 𝑏 ∈ 𝑌) ∧ 𝑐 = ⟨𝑎, 𝑏⟩) ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = 𝑎) ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑦) = 𝑏) → (𝐹‘𝑥) ≠ (𝐹‘𝑦))
42	6	ad8antr 772	. . . . . . . . . . . . . . . . 17 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) ∧ 𝑎 ∈ 𝑌) ∧ 𝑏 ∈ 𝑌) ∧ 𝑐 = ⟨𝑎, 𝑏⟩) ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = 𝑎) ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑦) = 𝑏) → 𝐹 Fn 𝑋)
43		qtophaus.e	. . . . . . . . . . . . . . . . . 18 ⊢ ∼ = (◡𝐹 ∘ 𝐹)
44	43	fcoinvbr 28799	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥 ∼ 𝑦 ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))
45	42, 24, 25, 44	syl3anc 1318	. . . . . . . . . . . . . . . 16 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) ∧ 𝑎 ∈ 𝑌) ∧ 𝑏 ∈ 𝑌) ∧ 𝑐 = ⟨𝑎, 𝑏⟩) ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = 𝑎) ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑦) = 𝑏) → (𝑥 ∼ 𝑦 ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))
46	45	necon3bbid 2819	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) ∧ 𝑎 ∈ 𝑌) ∧ 𝑏 ∈ 𝑌) ∧ 𝑐 = ⟨𝑎, 𝑏⟩) ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = 𝑎) ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑦) = 𝑏) → (¬ 𝑥 ∼ 𝑦 ↔ (𝐹‘𝑥) ≠ (𝐹‘𝑦)))
47	41, 46	mpbird 246	. . . . . . . . . . . . . 14 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) ∧ 𝑎 ∈ 𝑌) ∧ 𝑏 ∈ 𝑌) ∧ 𝑐 = ⟨𝑎, 𝑏⟩) ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = 𝑎) ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑦) = 𝑏) → ¬ 𝑥 ∼ 𝑦)
48		df-br 4584	. . . . . . . . . . . . . . 15 ⊢ (𝑥((𝑋 × 𝑋) ∖ ∼ )𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ((𝑋 × 𝑋) ∖ ∼ ))
49		brdif 4635	. . . . . . . . . . . . . . 15 ⊢ (𝑥((𝑋 × 𝑋) ∖ ∼ )𝑦 ↔ (𝑥(𝑋 × 𝑋)𝑦 ∧ ¬ 𝑥 ∼ 𝑦))
50	48, 49	bitr3i 265	. . . . . . . . . . . . . 14 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝑋 × 𝑋) ∖ ∼ ) ↔ (𝑥(𝑋 × 𝑋)𝑦 ∧ ¬ 𝑥 ∼ 𝑦))
51	29, 47, 50	sylanbrc 695	. . . . . . . . . . . . 13 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) ∧ 𝑎 ∈ 𝑌) ∧ 𝑏 ∈ 𝑌) ∧ 𝑐 = ⟨𝑎, 𝑏⟩) ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = 𝑎) ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑦) = 𝑏) → ⟨𝑥, 𝑦⟩ ∈ ((𝑋 × 𝑋) ∖ ∼ ))
52		qtophaus.h	. . . . . . . . . . . . . . 15 ⊢ 𝐻 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
53	52, 24, 25	fvproj 29227	. . . . . . . . . . . . . 14 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) ∧ 𝑎 ∈ 𝑌) ∧ 𝑏 ∈ 𝑌) ∧ 𝑐 = ⟨𝑎, 𝑏⟩) ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = 𝑎) ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑦) = 𝑏) → (𝐻‘⟨𝑥, 𝑦⟩) = ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
54	32, 53, 33	3eqtr4d 2654	. . . . . . . . . . . . 13 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) ∧ 𝑎 ∈ 𝑌) ∧ 𝑏 ∈ 𝑌) ∧ 𝑐 = ⟨𝑎, 𝑏⟩) ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = 𝑎) ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑦) = 𝑏) → (𝐻‘⟨𝑥, 𝑦⟩) = 𝑐)
55		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
56	55	eqeq1d 2612	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐻‘𝑧) = 𝑐 ↔ (𝐻‘⟨𝑥, 𝑦⟩) = 𝑐))
57	56	rspcev 3282	. . . . . . . . . . . . 13 ⊢ ((⟨𝑥, 𝑦⟩ ∈ ((𝑋 × 𝑋) ∖ ∼ ) ∧ (𝐻‘⟨𝑥, 𝑦⟩) = 𝑐) → ∃𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ )(𝐻‘𝑧) = 𝑐)
58	51, 54, 57	syl2anc 691	. . . . . . . . . . . 12 ⊢ (((((((((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) ∧ 𝑎 ∈ 𝑌) ∧ 𝑏 ∈ 𝑌) ∧ 𝑐 = ⟨𝑎, 𝑏⟩) ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = 𝑎) ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑦) = 𝑏) → ∃𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ )(𝐻‘𝑧) = 𝑐)
59		fofun 6029	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:𝑋–onto→𝑌 → Fun 𝐹)
60	4, 59	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → Fun 𝐹)
61	60	ad4antr 764	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) ∧ 𝑎 ∈ 𝑌) ∧ 𝑏 ∈ 𝑌) ∧ 𝑐 = ⟨𝑎, 𝑏⟩) → Fun 𝐹)
62	61	ad2antrr 758	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) ∧ 𝑎 ∈ 𝑌) ∧ 𝑏 ∈ 𝑌) ∧ 𝑐 = ⟨𝑎, 𝑏⟩) ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = 𝑎) → Fun 𝐹)
63		simp-4r 803	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) ∧ 𝑎 ∈ 𝑌) ∧ 𝑏 ∈ 𝑌) ∧ 𝑐 = ⟨𝑎, 𝑏⟩) ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = 𝑎) → 𝑏 ∈ 𝑌)
64		foima 6033	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝑋–onto→𝑌 → (𝐹 “ 𝑋) = 𝑌)
65	4, 64	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐹 “ 𝑋) = 𝑌)
66	65	ad4antr 764	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) ∧ 𝑎 ∈ 𝑌) ∧ 𝑏 ∈ 𝑌) ∧ 𝑐 = ⟨𝑎, 𝑏⟩) → (𝐹 “ 𝑋) = 𝑌)
67	66	ad2antrr 758	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) ∧ 𝑎 ∈ 𝑌) ∧ 𝑏 ∈ 𝑌) ∧ 𝑐 = ⟨𝑎, 𝑏⟩) ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = 𝑎) → (𝐹 “ 𝑋) = 𝑌)
68	63, 67	eleqtrrd 2691	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) ∧ 𝑎 ∈ 𝑌) ∧ 𝑏 ∈ 𝑌) ∧ 𝑐 = ⟨𝑎, 𝑏⟩) ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = 𝑎) → 𝑏 ∈ (𝐹 “ 𝑋))
69		fvelima 6158	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ 𝑏 ∈ (𝐹 “ 𝑋)) → ∃𝑦 ∈ 𝑋 (𝐹‘𝑦) = 𝑏)
70	62, 68, 69	syl2anc 691	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) ∧ 𝑎 ∈ 𝑌) ∧ 𝑏 ∈ 𝑌) ∧ 𝑐 = ⟨𝑎, 𝑏⟩) ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = 𝑎) → ∃𝑦 ∈ 𝑋 (𝐹‘𝑦) = 𝑏)
71	58, 70	r19.29a 3060	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) ∧ 𝑎 ∈ 𝑌) ∧ 𝑏 ∈ 𝑌) ∧ 𝑐 = ⟨𝑎, 𝑏⟩) ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = 𝑎) → ∃𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ )(𝐻‘𝑧) = 𝑐)
72		simpllr 795	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) ∧ 𝑎 ∈ 𝑌) ∧ 𝑏 ∈ 𝑌) ∧ 𝑐 = ⟨𝑎, 𝑏⟩) → 𝑎 ∈ 𝑌)
73	72, 66	eleqtrrd 2691	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) ∧ 𝑎 ∈ 𝑌) ∧ 𝑏 ∈ 𝑌) ∧ 𝑐 = ⟨𝑎, 𝑏⟩) → 𝑎 ∈ (𝐹 “ 𝑋))
74		fvelima 6158	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ 𝑎 ∈ (𝐹 “ 𝑋)) → ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑎)
75	61, 73, 74	syl2anc 691	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) ∧ 𝑎 ∈ 𝑌) ∧ 𝑏 ∈ 𝑌) ∧ 𝑐 = ⟨𝑎, 𝑏⟩) → ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑎)
76	71, 75	r19.29a 3060	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) ∧ 𝑎 ∈ 𝑌) ∧ 𝑏 ∈ 𝑌) ∧ 𝑐 = ⟨𝑎, 𝑏⟩) → ∃𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ )(𝐻‘𝑧) = 𝑐)
77		simpr 476	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) → 𝑐 ∈ ((𝑌 × 𝑌) ∖ I ))
78	77	eldifad 3552	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) → 𝑐 ∈ (𝑌 × 𝑌))
79		elxp2 5056	. . . . . . . . . . 11 ⊢ (𝑐 ∈ (𝑌 × 𝑌) ↔ ∃𝑎 ∈ 𝑌 ∃𝑏 ∈ 𝑌 𝑐 = ⟨𝑎, 𝑏⟩)
80	78, 79	sylib 207	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) → ∃𝑎 ∈ 𝑌 ∃𝑏 ∈ 𝑌 𝑐 = ⟨𝑎, 𝑏⟩)
81	76, 80	r19.29vva 3062	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )) → ∃𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ )(𝐻‘𝑧) = 𝑐)
82		simpr 476	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ )) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑧 = ⟨𝑥, 𝑦⟩)
83	82	fveq2d 6107	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ )) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐻‘𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
84		simp-4r 803	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ )) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐻‘𝑧) = 𝑐)
85		simpllr 795	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ )) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑥 ∈ 𝑋)
86		simplr 788	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ )) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑦 ∈ 𝑋)
87	52, 85, 86	fvproj 29227	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ )) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐻‘⟨𝑥, 𝑦⟩) = ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
88	83, 84, 87	3eqtr3d 2652	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ )) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑐 = ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
89		fof 6028	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌)
90	4, 89	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌)
91	90	ad5antr 766	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ )) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝐹:𝑋⟶𝑌)
92	91, 85	ffvelrnd 6268	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ )) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹‘𝑥) ∈ 𝑌)
93	91, 86	ffvelrnd 6268	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ )) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹‘𝑦) ∈ 𝑌)
94		opelxp 5070	. . . . . . . . . . . . . 14 ⊢ (⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ ∈ (𝑌 × 𝑌) ↔ ((𝐹‘𝑥) ∈ 𝑌 ∧ (𝐹‘𝑦) ∈ 𝑌))
95	92, 93, 94	sylanbrc 695	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ )) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ ∈ (𝑌 × 𝑌))
96		simp-5r 805	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ )) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ ))
97	82, 96	eqeltrrd 2689	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ )) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → ⟨𝑥, 𝑦⟩ ∈ ((𝑋 × 𝑋) ∖ ∼ ))
98	50	simprbi 479	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝑋 × 𝑋) ∖ ∼ ) → ¬ 𝑥 ∼ 𝑦)
99	97, 98	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ )) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → ¬ 𝑥 ∼ 𝑦)
100	6	ad5antr 766	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ )) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝐹 Fn 𝑋)
101	100, 85, 86, 44	syl3anc 1318	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ )) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝑥 ∼ 𝑦 ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))
102	101	necon3bbid 2819	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ )) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (¬ 𝑥 ∼ 𝑦 ↔ (𝐹‘𝑥) ≠ (𝐹‘𝑦)))
103	99, 102	mpbid 221	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ )) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝐹‘𝑥) ≠ (𝐹‘𝑦))
104	95, 103, 39	sylanbrc 695	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ )) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ ∈ ((𝑌 × 𝑌) ∖ I ))
105	88, 104	eqeltrd 2688	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ )) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → 𝑐 ∈ ((𝑌 × 𝑌) ∖ I ))
106		eldifi 3694	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ ) → 𝑧 ∈ (𝑋 × 𝑋))
107	106	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ )) → 𝑧 ∈ (𝑋 × 𝑋))
108		elxp2 5056	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (𝑋 × 𝑋) ↔ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 𝑧 = ⟨𝑥, 𝑦⟩)
109	107, 108	sylib 207	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ )) → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 𝑧 = ⟨𝑥, 𝑦⟩)
110	109	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ )) ∧ (𝐻‘𝑧) = 𝑐) → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 𝑧 = ⟨𝑥, 𝑦⟩)
111	105, 110	r19.29vva 3062	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ )) ∧ (𝐻‘𝑧) = 𝑐) → 𝑐 ∈ ((𝑌 × 𝑌) ∖ I ))
112	111	r19.29an 3059	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∃𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ )(𝐻‘𝑧) = 𝑐) → 𝑐 ∈ ((𝑌 × 𝑌) ∖ I ))
113	81, 112	impbida 873	. . . . . . . 8 ⊢ (𝜑 → (𝑐 ∈ ((𝑌 × 𝑌) ∖ I ) ↔ ∃𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ )(𝐻‘𝑧) = 𝑐))
114		opex 4859	. . . . . . . . . 10 ⊢ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ ∈ V
115	52, 114	fnmpt2i 7128	. . . . . . . . 9 ⊢ 𝐻 Fn (𝑋 × 𝑋)
116		difss 3699	. . . . . . . . 9 ⊢ ((𝑋 × 𝑋) ∖ ∼ ) ⊆ (𝑋 × 𝑋)
117		fvelimab 6163	. . . . . . . . 9 ⊢ ((𝐻 Fn (𝑋 × 𝑋) ∧ ((𝑋 × 𝑋) ∖ ∼ ) ⊆ (𝑋 × 𝑋)) → (𝑐 ∈ (𝐻 “ ((𝑋 × 𝑋) ∖ ∼ )) ↔ ∃𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ )(𝐻‘𝑧) = 𝑐))
118	115, 116, 117	mp2an 704	. . . . . . . 8 ⊢ (𝑐 ∈ (𝐻 “ ((𝑋 × 𝑋) ∖ ∼ )) ↔ ∃𝑧 ∈ ((𝑋 × 𝑋) ∖ ∼ )(𝐻‘𝑧) = 𝑐)
119	113, 118	syl6rbbr 278	. . . . . . 7 ⊢ (𝜑 → (𝑐 ∈ (𝐻 “ ((𝑋 × 𝑋) ∖ ∼ )) ↔ 𝑐 ∈ ((𝑌 × 𝑌) ∖ I )))
120	119	eqrdv 2608	. . . . . 6 ⊢ (𝜑 → (𝐻 “ ((𝑋 × 𝑋) ∖ ∼ )) = ((𝑌 × 𝑌) ∖ I ))
121		ssv 3588	. . . . . . 7 ⊢ 𝑌 ⊆ V
122		xpss2 5152	. . . . . . 7 ⊢ (𝑌 ⊆ V → (𝑌 × 𝑌) ⊆ (𝑌 × V))
123		difres 28795	. . . . . . 7 ⊢ ((𝑌 × 𝑌) ⊆ (𝑌 × V) → ((𝑌 × 𝑌) ∖ ( I ↾ 𝑌)) = ((𝑌 × 𝑌) ∖ I ))
124	121, 122, 123	mp2b 10	. . . . . 6 ⊢ ((𝑌 × 𝑌) ∖ ( I ↾ 𝑌)) = ((𝑌 × 𝑌) ∖ I )
125	120, 124	syl6eqr 2662	. . . . 5 ⊢ (𝜑 → (𝐻 “ ((𝑋 × 𝑋) ∖ ∼ )) = ((𝑌 × 𝑌) ∖ ( I ↾ 𝑌)))
126	7	toptopon 20548	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
127	3, 126	sylib 207	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
128		qtoptopon 21317	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
129	127, 4, 128	syl2anc 691	. . . . . 6 ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
130		qtophaus.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹))
131	130	ralrimiva 2949	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝐽 (𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹))
132		imaeq2 5381	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝐹 “ 𝑥) = (𝐹 “ 𝑦))
133	132	eleq1d 2672	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹) ↔ (𝐹 “ 𝑦) ∈ (𝐽 qTop 𝐹)))
134	133	cbvralv 3147	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐽 (𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹) ↔ ∀𝑦 ∈ 𝐽 (𝐹 “ 𝑦) ∈ (𝐽 qTop 𝐹))
135	131, 134	sylib 207	. . . . . . 7 ⊢ (𝜑 → ∀𝑦 ∈ 𝐽 (𝐹 “ 𝑦) ∈ (𝐽 qTop 𝐹))
136	135	r19.21bi 2916	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → (𝐹 “ 𝑦) ∈ (𝐽 qTop 𝐹))
137	7, 7	txuni 21205	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → (𝑋 × 𝑋) = ∪ (𝐽 ×t 𝐽))
138	3, 3, 137	syl2anc 691	. . . . . . . 8 ⊢ (𝜑 → (𝑋 × 𝑋) = ∪ (𝐽 ×t 𝐽))
139	138	difeq1d 3689	. . . . . . 7 ⊢ (𝜑 → ((𝑋 × 𝑋) ∖ ∼ ) = (∪ (𝐽 ×t 𝐽) ∖ ∼ ))
140		qtophaus.4	. . . . . . . 8 ⊢ (𝜑 → ∼ ∈ (Clsd‘(𝐽 ×t 𝐽)))
141		txtop 21182	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → (𝐽 ×t 𝐽) ∈ Top)
142	3, 3, 141	syl2anc 691	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 ×t 𝐽) ∈ Top)
143		fcoinver 28798	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn 𝑋 → (◡𝐹 ∘ 𝐹) Er 𝑋)
144	6, 143	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (◡𝐹 ∘ 𝐹) Er 𝑋)
145		ereq1 7636	. . . . . . . . . . . . 13 ⊢ ( ∼ = (◡𝐹 ∘ 𝐹) → ( ∼ Er 𝑋 ↔ (◡𝐹 ∘ 𝐹) Er 𝑋))
146	43, 145	ax-mp 5	. . . . . . . . . . . 12 ⊢ ( ∼ Er 𝑋 ↔ (◡𝐹 ∘ 𝐹) Er 𝑋)
147	144, 146	sylibr 223	. . . . . . . . . . 11 ⊢ (𝜑 → ∼ Er 𝑋)
148		erssxp 7652	. . . . . . . . . . 11 ⊢ ( ∼ Er 𝑋 → ∼ ⊆ (𝑋 × 𝑋))
149	147, 148	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ∼ ⊆ (𝑋 × 𝑋))
150	149, 138	sseqtrd 3604	. . . . . . . . 9 ⊢ (𝜑 → ∼ ⊆ ∪ (𝐽 ×t 𝐽))
151		eqid 2610	. . . . . . . . . 10 ⊢ ∪ (𝐽 ×t 𝐽) = ∪ (𝐽 ×t 𝐽)
152	151	iscld2 20642	. . . . . . . . 9 ⊢ (((𝐽 ×t 𝐽) ∈ Top ∧ ∼ ⊆ ∪ (𝐽 ×t 𝐽)) → ( ∼ ∈ (Clsd‘(𝐽 ×t 𝐽)) ↔ (∪ (𝐽 ×t 𝐽) ∖ ∼ ) ∈ (𝐽 ×t 𝐽)))
153	142, 150, 152	syl2anc 691	. . . . . . . 8 ⊢ (𝜑 → ( ∼ ∈ (Clsd‘(𝐽 ×t 𝐽)) ↔ (∪ (𝐽 ×t 𝐽) ∖ ∼ ) ∈ (𝐽 ×t 𝐽)))
154	140, 153	mpbid 221	. . . . . . 7 ⊢ (𝜑 → (∪ (𝐽 ×t 𝐽) ∖ ∼ ) ∈ (𝐽 ×t 𝐽))
155	139, 154	eqeltrd 2688	. . . . . 6 ⊢ (𝜑 → ((𝑋 × 𝑋) ∖ ∼ ) ∈ (𝐽 ×t 𝐽))
156	90, 90, 127, 127, 129, 129, 130, 136, 155, 52	txomap 29229	. . . . 5 ⊢ (𝜑 → (𝐻 “ ((𝑋 × 𝑋) ∖ ∼ )) ∈ ((𝐽 qTop 𝐹) ×t (𝐽 qTop 𝐹)))
157	125, 156	eqeltrrd 2689	. . . 4 ⊢ (𝜑 → ((𝑌 × 𝑌) ∖ ( I ↾ 𝑌)) ∈ ((𝐽 qTop 𝐹) ×t (𝐽 qTop 𝐹)))
158	23, 157	eqeltrd 2688	. . 3 ⊢ (𝜑 → (∪ ((𝐽 qTop 𝐹) ×t (𝐽 qTop 𝐹)) ∖ ( I ↾ ∪ (𝐽 qTop 𝐹))) ∈ ((𝐽 qTop 𝐹) ×t (𝐽 qTop 𝐹)))
159		eqid 2610	. . . . 5 ⊢ ∪ ((𝐽 qTop 𝐹) ×t (𝐽 qTop 𝐹)) = ∪ ((𝐽 qTop 𝐹) ×t (𝐽 qTop 𝐹))
160	159	iscld2 20642	. . . 4 ⊢ ((((𝐽 qTop 𝐹) ×t (𝐽 qTop 𝐹)) ∈ Top ∧ ( I ↾ ∪ (𝐽 qTop 𝐹)) ⊆ ∪ ((𝐽 qTop 𝐹) ×t (𝐽 qTop 𝐹))) → (( I ↾ ∪ (𝐽 qTop 𝐹)) ∈ (Clsd‘((𝐽 qTop 𝐹) ×t (𝐽 qTop 𝐹))) ↔ (∪ ((𝐽 qTop 𝐹) ×t (𝐽 qTop 𝐹)) ∖ ( I ↾ ∪ (𝐽 qTop 𝐹))) ∈ ((𝐽 qTop 𝐹) ×t (𝐽 qTop 𝐹))))
161	160	biimpar 501	. . 3 ⊢ (((((𝐽 qTop 𝐹) ×t (𝐽 qTop 𝐹)) ∈ Top ∧ ( I ↾ ∪ (𝐽 qTop 𝐹)) ⊆ ∪ ((𝐽 qTop 𝐹) ×t (𝐽 qTop 𝐹))) ∧ (∪ ((𝐽 qTop 𝐹) ×t (𝐽 qTop 𝐹)) ∖ ( I ↾ ∪ (𝐽 qTop 𝐹))) ∈ ((𝐽 qTop 𝐹) ×t (𝐽 qTop 𝐹))) → ( I ↾ ∪ (𝐽 qTop 𝐹)) ∈ (Clsd‘((𝐽 qTop 𝐹) ×t (𝐽 qTop 𝐹))))
162	11, 16, 158, 161	syl21anc 1317	. 2 ⊢ (𝜑 → ( I ↾ ∪ (𝐽 qTop 𝐹)) ∈ (Clsd‘((𝐽 qTop 𝐹) ×t (𝐽 qTop 𝐹))))
163	13	hausdiag 21258	. 2 ⊢ ((𝐽 qTop 𝐹) ∈ Haus ↔ ((𝐽 qTop 𝐹) ∈ Top ∧ ( I ↾ ∪ (𝐽 qTop 𝐹)) ∈ (Clsd‘((𝐽 qTop 𝐹) ×t (𝐽 qTop 𝐹)))))
164	9, 162, 163	sylanbrc 695	1 ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ Haus)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ⟨cop 4131  ∪ cuni 4372   class class class wbr 4583   I cid 4948   × cxp 5036  ^◡ccnv 5037   ↾ cres 5040   “ cima 5041   ∘ ccom 5042  Rel wrel 5043  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   Er wer 7626   qTop cqtop 15986  Topctop 20517  TopOnctopon 20518  Clsdccld 20630  Hauscha 20922   ×_t ctx 21173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-er 7629  df-topgen 15927  df-qtop 15990  df-top 20521  df-bases 20522  df-topon 20523  df-cld 20633  df-haus 20929  df-tx 21175

This theorem is referenced by: (None)