Theorem xkohaus 22255

Description: If the codomain space is Hausdorff, then the compact-open topology of continuous functions is also Hausdorff. (Contributed by Mario Carneiro, 19-Mar-2015.)

Assertion

Ref	Expression
xkohaus	⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) → (𝑆 ↑ko 𝑅) ∈ Haus)

Proof of Theorem xkohaus

Dummy variables 𝑎 𝑏 𝑓 𝑔 ℎ 𝑢 𝑣 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		haustop 21933	. . 3 ⊢ (𝑆 ∈ Haus → 𝑆 ∈ Top)
2		xkotop 22190	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝑅) ∈ Top)
3	1, 2	sylan2 594	. 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) → (𝑆 ↑ko 𝑅) ∈ Top)
4		eqid 2821	. . . . . . . 8 ⊢ (𝑆 ↑ko 𝑅) = (𝑆 ↑ko 𝑅)
5	4	xkouni 22201	. . . . . . 7 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) = ∪ (𝑆 ↑ko 𝑅))
6	1, 5	sylan2 594	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) → (𝑅 Cn 𝑆) = ∪ (𝑆 ↑ko 𝑅))
7	6	eleq2d 2898	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) → (𝑓 ∈ (𝑅 Cn 𝑆) ↔ 𝑓 ∈ ∪ (𝑆 ↑ko 𝑅)))
8	6	eleq2d 2898	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) → (𝑔 ∈ (𝑅 Cn 𝑆) ↔ 𝑔 ∈ ∪ (𝑆 ↑ko 𝑅)))
9	7, 8	anbi12d 632	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) → ((𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) ↔ (𝑓 ∈ ∪ (𝑆 ↑ko 𝑅) ∧ 𝑔 ∈ ∪ (𝑆 ↑ko 𝑅))))
10		simprl 769	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → 𝑓 ∈ (𝑅 Cn 𝑆))
11		eqid 2821	. . . . . . . . . . 11 ⊢ ∪ 𝑅 = ∪ 𝑅
12		eqid 2821	. . . . . . . . . . 11 ⊢ ∪ 𝑆 = ∪ 𝑆
13	11, 12	cnf 21848	. . . . . . . . . 10 ⊢ (𝑓 ∈ (𝑅 Cn 𝑆) → 𝑓:∪ 𝑅⟶∪ 𝑆)
14	10, 13	syl 17	. . . . . . . . 9 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → 𝑓:∪ 𝑅⟶∪ 𝑆)
15	14	ffnd 6509	. . . . . . . 8 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → 𝑓 Fn ∪ 𝑅)
16		simprr 771	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → 𝑔 ∈ (𝑅 Cn 𝑆))
17	11, 12	cnf 21848	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝑅 Cn 𝑆) → 𝑔:∪ 𝑅⟶∪ 𝑆)
18	16, 17	syl 17	. . . . . . . . 9 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → 𝑔:∪ 𝑅⟶∪ 𝑆)
19	18	ffnd 6509	. . . . . . . 8 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → 𝑔 Fn ∪ 𝑅)
20		eqfnfv 6796	. . . . . . . 8 ⊢ ((𝑓 Fn ∪ 𝑅 ∧ 𝑔 Fn ∪ 𝑅) → (𝑓 = 𝑔 ↔ ∀𝑥 ∈ ∪ 𝑅(𝑓‘𝑥) = (𝑔‘𝑥)))
21	15, 19, 20	syl2anc 586	. . . . . . 7 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → (𝑓 = 𝑔 ↔ ∀𝑥 ∈ ∪ 𝑅(𝑓‘𝑥) = (𝑔‘𝑥)))
22	21	necon3abid 3052	. . . . . 6 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → (𝑓 ≠ 𝑔 ↔ ¬ ∀𝑥 ∈ ∪ 𝑅(𝑓‘𝑥) = (𝑔‘𝑥)))
23		rexnal 3238	. . . . . . 7 ⊢ (∃𝑥 ∈ ∪ 𝑅 ¬ (𝑓‘𝑥) = (𝑔‘𝑥) ↔ ¬ ∀𝑥 ∈ ∪ 𝑅(𝑓‘𝑥) = (𝑔‘𝑥))
24		df-ne 3017	. . . . . . . . . 10 ⊢ ((𝑓‘𝑥) ≠ (𝑔‘𝑥) ↔ ¬ (𝑓‘𝑥) = (𝑔‘𝑥))
25		simpllr 774	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ (𝑥 ∈ ∪ 𝑅 ∧ (𝑓‘𝑥) ≠ (𝑔‘𝑥))) → 𝑆 ∈ Haus)
26	14	adantr 483	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ (𝑥 ∈ ∪ 𝑅 ∧ (𝑓‘𝑥) ≠ (𝑔‘𝑥))) → 𝑓:∪ 𝑅⟶∪ 𝑆)
27		simprl 769	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ (𝑥 ∈ ∪ 𝑅 ∧ (𝑓‘𝑥) ≠ (𝑔‘𝑥))) → 𝑥 ∈ ∪ 𝑅)
28	26, 27	ffvelrnd 6846	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ (𝑥 ∈ ∪ 𝑅 ∧ (𝑓‘𝑥) ≠ (𝑔‘𝑥))) → (𝑓‘𝑥) ∈ ∪ 𝑆)
29	18	adantr 483	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ (𝑥 ∈ ∪ 𝑅 ∧ (𝑓‘𝑥) ≠ (𝑔‘𝑥))) → 𝑔:∪ 𝑅⟶∪ 𝑆)
30	29, 27	ffvelrnd 6846	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ (𝑥 ∈ ∪ 𝑅 ∧ (𝑓‘𝑥) ≠ (𝑔‘𝑥))) → (𝑔‘𝑥) ∈ ∪ 𝑆)
31		simprr 771	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ (𝑥 ∈ ∪ 𝑅 ∧ (𝑓‘𝑥) ≠ (𝑔‘𝑥))) → (𝑓‘𝑥) ≠ (𝑔‘𝑥))
32	12	hausnei 21930	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Haus ∧ ((𝑓‘𝑥) ∈ ∪ 𝑆 ∧ (𝑔‘𝑥) ∈ ∪ 𝑆 ∧ (𝑓‘𝑥) ≠ (𝑔‘𝑥))) → ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑆 ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))
33	25, 28, 30, 31, 32	syl13anc 1368	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ (𝑥 ∈ ∪ 𝑅 ∧ (𝑓‘𝑥) ≠ (𝑔‘𝑥))) → ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑆 ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))
34	33	expr 459	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) → ((𝑓‘𝑥) ≠ (𝑔‘𝑥) → ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑆 ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅)))
35	24, 34	syl5bir 245	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) → (¬ (𝑓‘𝑥) = (𝑔‘𝑥) → ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑆 ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅)))
36		simp-4l 781	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → 𝑅 ∈ Top)
37	1	ad4antlr 731	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → 𝑆 ∈ Top)
38		simplr 767	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → 𝑥 ∈ ∪ 𝑅)
39	38	snssd 4735	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → {𝑥} ⊆ ∪ 𝑅)
40		toptopon2 21520	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘∪ 𝑅))
41	36, 40	sylib 220	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → 𝑅 ∈ (TopOn‘∪ 𝑅))
42		restsn2 21773	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ (TopOn‘∪ 𝑅) ∧ 𝑥 ∈ ∪ 𝑅) → (𝑅 ↾t {𝑥}) = 𝒫 {𝑥})
43	41, 38, 42	syl2anc 586	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → (𝑅 ↾t {𝑥}) = 𝒫 {𝑥})
44		snfi 8588	. . . . . . . . . . . . . . 15 ⊢ {𝑥} ∈ Fin
45		discmp 22000	. . . . . . . . . . . . . . 15 ⊢ ({𝑥} ∈ Fin ↔ 𝒫 {𝑥} ∈ Comp)
46	44, 45	mpbi 232	. . . . . . . . . . . . . 14 ⊢ 𝒫 {𝑥} ∈ Comp
47	43, 46	eqeltrdi 2921	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → (𝑅 ↾t {𝑥}) ∈ Comp)
48		simprll 777	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → 𝑎 ∈ 𝑆)
49	11, 36, 37, 39, 47, 48	xkoopn 22191	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∈ (𝑆 ↑ko 𝑅))
50		simprlr 778	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → 𝑏 ∈ 𝑆)
51	11, 36, 37, 39, 47, 50	xkoopn 22191	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑏} ∈ (𝑆 ↑ko 𝑅))
52		imaeq1 5918	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝑓 → (ℎ “ {𝑥}) = (𝑓 “ {𝑥}))
53	52	sseq1d 3997	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑓 → ((ℎ “ {𝑥}) ⊆ 𝑎 ↔ (𝑓 “ {𝑥}) ⊆ 𝑎))
54	10	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → 𝑓 ∈ (𝑅 Cn 𝑆))
55	15	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → 𝑓 Fn ∪ 𝑅)
56		fnsnfv 6737	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 Fn ∪ 𝑅 ∧ 𝑥 ∈ ∪ 𝑅) → {(𝑓‘𝑥)} = (𝑓 “ {𝑥}))
57	55, 38, 56	syl2anc 586	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → {(𝑓‘𝑥)} = (𝑓 “ {𝑥}))
58		simprr1 1217	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → (𝑓‘𝑥) ∈ 𝑎)
59	58	snssd 4735	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → {(𝑓‘𝑥)} ⊆ 𝑎)
60	57, 59	eqsstrrd 4005	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → (𝑓 “ {𝑥}) ⊆ 𝑎)
61	53, 54, 60	elrabd 3681	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → 𝑓 ∈ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎})
62		imaeq1 5918	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝑔 → (ℎ “ {𝑥}) = (𝑔 “ {𝑥}))
63	62	sseq1d 3997	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑔 → ((ℎ “ {𝑥}) ⊆ 𝑏 ↔ (𝑔 “ {𝑥}) ⊆ 𝑏))
64	16	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → 𝑔 ∈ (𝑅 Cn 𝑆))
65	19	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → 𝑔 Fn ∪ 𝑅)
66		fnsnfv 6737	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 Fn ∪ 𝑅 ∧ 𝑥 ∈ ∪ 𝑅) → {(𝑔‘𝑥)} = (𝑔 “ {𝑥}))
67	65, 38, 66	syl2anc 586	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → {(𝑔‘𝑥)} = (𝑔 “ {𝑥}))
68		simprr2 1218	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → (𝑔‘𝑥) ∈ 𝑏)
69	68	snssd 4735	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → {(𝑔‘𝑥)} ⊆ 𝑏)
70	67, 69	eqsstrrd 4005	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → (𝑔 “ {𝑥}) ⊆ 𝑏)
71	63, 64, 70	elrabd 3681	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → 𝑔 ∈ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑏})
72		inrab 4274	. . . . . . . . . . . . 13 ⊢ ({ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∩ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑏}) = {ℎ ∈ (𝑅 Cn 𝑆) ∣ ((ℎ “ {𝑥}) ⊆ 𝑎 ∧ (ℎ “ {𝑥}) ⊆ 𝑏)}
73		simpllr 774	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) ∧ ℎ ∈ (𝑅 Cn 𝑆)) → 𝑥 ∈ ∪ 𝑅)
74	11, 12	cnf 21848	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℎ ∈ (𝑅 Cn 𝑆) → ℎ:∪ 𝑅⟶∪ 𝑆)
75	74	fdmd 6517	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℎ ∈ (𝑅 Cn 𝑆) → dom ℎ = ∪ 𝑅)
76	75	adantl 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) ∧ ℎ ∈ (𝑅 Cn 𝑆)) → dom ℎ = ∪ 𝑅)
77	73, 76	eleqtrrd 2916	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) ∧ ℎ ∈ (𝑅 Cn 𝑆)) → 𝑥 ∈ dom ℎ)
78		simprr3 1219	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → (𝑎 ∩ 𝑏) = ∅)
79	78	adantr 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) ∧ ℎ ∈ (𝑅 Cn 𝑆)) → (𝑎 ∩ 𝑏) = ∅)
80		sseq0 4352	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((ℎ “ {𝑥}) ⊆ (𝑎 ∩ 𝑏) ∧ (𝑎 ∩ 𝑏) = ∅) → (ℎ “ {𝑥}) = ∅)
81	80	expcom 416	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ∩ 𝑏) = ∅ → ((ℎ “ {𝑥}) ⊆ (𝑎 ∩ 𝑏) → (ℎ “ {𝑥}) = ∅))
82	79, 81	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) ∧ ℎ ∈ (𝑅 Cn 𝑆)) → ((ℎ “ {𝑥}) ⊆ (𝑎 ∩ 𝑏) → (ℎ “ {𝑥}) = ∅))
83		imadisj 5942	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℎ “ {𝑥}) = ∅ ↔ (dom ℎ ∩ {𝑥}) = ∅)
84		disjsn 4640	. . . . . . . . . . . . . . . . . . 19 ⊢ ((dom ℎ ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ dom ℎ)
85	83, 84	bitri 277	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℎ “ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ dom ℎ)
86	82, 85	syl6ib 253	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) ∧ ℎ ∈ (𝑅 Cn 𝑆)) → ((ℎ “ {𝑥}) ⊆ (𝑎 ∩ 𝑏) → ¬ 𝑥 ∈ dom ℎ))
87	77, 86	mt2d 138	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) ∧ ℎ ∈ (𝑅 Cn 𝑆)) → ¬ (ℎ “ {𝑥}) ⊆ (𝑎 ∩ 𝑏))
88		ssin 4206	. . . . . . . . . . . . . . . 16 ⊢ (((ℎ “ {𝑥}) ⊆ 𝑎 ∧ (ℎ “ {𝑥}) ⊆ 𝑏) ↔ (ℎ “ {𝑥}) ⊆ (𝑎 ∩ 𝑏))
89	87, 88	sylnibr 331	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) ∧ ℎ ∈ (𝑅 Cn 𝑆)) → ¬ ((ℎ “ {𝑥}) ⊆ 𝑎 ∧ (ℎ “ {𝑥}) ⊆ 𝑏))
90	89	ralrimiva 3182	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → ∀ℎ ∈ (𝑅 Cn 𝑆) ¬ ((ℎ “ {𝑥}) ⊆ 𝑎 ∧ (ℎ “ {𝑥}) ⊆ 𝑏))
91		rabeq0 4337	. . . . . . . . . . . . . 14 ⊢ ({ℎ ∈ (𝑅 Cn 𝑆) ∣ ((ℎ “ {𝑥}) ⊆ 𝑎 ∧ (ℎ “ {𝑥}) ⊆ 𝑏)} = ∅ ↔ ∀ℎ ∈ (𝑅 Cn 𝑆) ¬ ((ℎ “ {𝑥}) ⊆ 𝑎 ∧ (ℎ “ {𝑥}) ⊆ 𝑏))
92	90, 91	sylibr 236	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → {ℎ ∈ (𝑅 Cn 𝑆) ∣ ((ℎ “ {𝑥}) ⊆ 𝑎 ∧ (ℎ “ {𝑥}) ⊆ 𝑏)} = ∅)
93	72, 92	syl5eq 2868	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → ({ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∩ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑏}) = ∅)
94		eleq2 2901	. . . . . . . . . . . . . 14 ⊢ (𝑢 = {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} → (𝑓 ∈ 𝑢 ↔ 𝑓 ∈ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎}))
95		ineq1 4180	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} → (𝑢 ∩ 𝑣) = ({ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∩ 𝑣))
96	95	eqeq1d 2823	. . . . . . . . . . . . . 14 ⊢ (𝑢 = {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} → ((𝑢 ∩ 𝑣) = ∅ ↔ ({ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∩ 𝑣) = ∅))
97	94, 96	3anbi13d 1434	. . . . . . . . . . . . 13 ⊢ (𝑢 = {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} → ((𝑓 ∈ 𝑢 ∧ 𝑔 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅) ↔ (𝑓 ∈ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∧ 𝑔 ∈ 𝑣 ∧ ({ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∩ 𝑣) = ∅)))
98		eleq2 2901	. . . . . . . . . . . . . 14 ⊢ (𝑣 = {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑏} → (𝑔 ∈ 𝑣 ↔ 𝑔 ∈ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑏}))
99		ineq2 4182	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑏} → ({ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∩ 𝑣) = ({ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∩ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑏}))
100	99	eqeq1d 2823	. . . . . . . . . . . . . 14 ⊢ (𝑣 = {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑏} → (({ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∩ 𝑣) = ∅ ↔ ({ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∩ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑏}) = ∅))
101	98, 100	3anbi23d 1435	. . . . . . . . . . . . 13 ⊢ (𝑣 = {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑏} → ((𝑓 ∈ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∧ 𝑔 ∈ 𝑣 ∧ ({ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∩ 𝑣) = ∅) ↔ (𝑓 ∈ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∧ 𝑔 ∈ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑏} ∧ ({ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∩ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑏}) = ∅)))
102	97, 101	rspc2ev 3634	. . . . . . . . . . . 12 ⊢ (({ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∈ (𝑆 ↑ko 𝑅) ∧ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑏} ∈ (𝑆 ↑ko 𝑅) ∧ (𝑓 ∈ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∧ 𝑔 ∈ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑏} ∧ ({ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑎} ∩ {ℎ ∈ (𝑅 Cn 𝑆) ∣ (ℎ “ {𝑥}) ⊆ 𝑏}) = ∅)) → ∃𝑢 ∈ (𝑆 ↑ko 𝑅)∃𝑣 ∈ (𝑆 ↑ko 𝑅)(𝑓 ∈ 𝑢 ∧ 𝑔 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))
103	49, 51, 61, 71, 93, 102	syl113anc 1378	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅))) → ∃𝑢 ∈ (𝑆 ↑ko 𝑅)∃𝑣 ∈ (𝑆 ↑ko 𝑅)(𝑓 ∈ 𝑢 ∧ 𝑔 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))
104	103	expr 459	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) → (((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅) → ∃𝑢 ∈ (𝑆 ↑ko 𝑅)∃𝑣 ∈ (𝑆 ↑ko 𝑅)(𝑓 ∈ 𝑢 ∧ 𝑔 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅)))
105	104	rexlimdvva 3294	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) → (∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑆 ((𝑓‘𝑥) ∈ 𝑎 ∧ (𝑔‘𝑥) ∈ 𝑏 ∧ (𝑎 ∩ 𝑏) = ∅) → ∃𝑢 ∈ (𝑆 ↑ko 𝑅)∃𝑣 ∈ (𝑆 ↑ko 𝑅)(𝑓 ∈ 𝑢 ∧ 𝑔 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅)))
106	35, 105	syld 47	. . . . . . . 8 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) ∧ 𝑥 ∈ ∪ 𝑅) → (¬ (𝑓‘𝑥) = (𝑔‘𝑥) → ∃𝑢 ∈ (𝑆 ↑ko 𝑅)∃𝑣 ∈ (𝑆 ↑ko 𝑅)(𝑓 ∈ 𝑢 ∧ 𝑔 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅)))
107	106	rexlimdva 3284	. . . . . . 7 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → (∃𝑥 ∈ ∪ 𝑅 ¬ (𝑓‘𝑥) = (𝑔‘𝑥) → ∃𝑢 ∈ (𝑆 ↑ko 𝑅)∃𝑣 ∈ (𝑆 ↑ko 𝑅)(𝑓 ∈ 𝑢 ∧ 𝑔 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅)))
108	23, 107	syl5bir 245	. . . . . 6 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → (¬ ∀𝑥 ∈ ∪ 𝑅(𝑓‘𝑥) = (𝑔‘𝑥) → ∃𝑢 ∈ (𝑆 ↑ko 𝑅)∃𝑣 ∈ (𝑆 ↑ko 𝑅)(𝑓 ∈ 𝑢 ∧ 𝑔 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅)))
109	22, 108	sylbid 242	. . . . 5 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) ∧ (𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆))) → (𝑓 ≠ 𝑔 → ∃𝑢 ∈ (𝑆 ↑ko 𝑅)∃𝑣 ∈ (𝑆 ↑ko 𝑅)(𝑓 ∈ 𝑢 ∧ 𝑔 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅)))
110	109	ex 415	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) → ((𝑓 ∈ (𝑅 Cn 𝑆) ∧ 𝑔 ∈ (𝑅 Cn 𝑆)) → (𝑓 ≠ 𝑔 → ∃𝑢 ∈ (𝑆 ↑ko 𝑅)∃𝑣 ∈ (𝑆 ↑ko 𝑅)(𝑓 ∈ 𝑢 ∧ 𝑔 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))))
111	9, 110	sylbird 262	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) → ((𝑓 ∈ ∪ (𝑆 ↑ko 𝑅) ∧ 𝑔 ∈ ∪ (𝑆 ↑ko 𝑅)) → (𝑓 ≠ 𝑔 → ∃𝑢 ∈ (𝑆 ↑ko 𝑅)∃𝑣 ∈ (𝑆 ↑ko 𝑅)(𝑓 ∈ 𝑢 ∧ 𝑔 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))))
112	111	ralrimivv 3190	. 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) → ∀𝑓 ∈ ∪ (𝑆 ↑ko 𝑅)∀𝑔 ∈ ∪ (𝑆 ↑ko 𝑅)(𝑓 ≠ 𝑔 → ∃𝑢 ∈ (𝑆 ↑ko 𝑅)∃𝑣 ∈ (𝑆 ↑ko 𝑅)(𝑓 ∈ 𝑢 ∧ 𝑔 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅)))
113		eqid 2821	. . 3 ⊢ ∪ (𝑆 ↑ko 𝑅) = ∪ (𝑆 ↑ko 𝑅)
114	113	ishaus 21924	. 2 ⊢ ((𝑆 ↑ko 𝑅) ∈ Haus ↔ ((𝑆 ↑ko 𝑅) ∈ Top ∧ ∀𝑓 ∈ ∪ (𝑆 ↑ko 𝑅)∀𝑔 ∈ ∪ (𝑆 ↑ko 𝑅)(𝑓 ≠ 𝑔 → ∃𝑢 ∈ (𝑆 ↑ko 𝑅)∃𝑣 ∈ (𝑆 ↑ko 𝑅)(𝑓 ∈ 𝑢 ∧ 𝑔 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))))
115	3, 112, 114	sylanbrc 585	1 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) → (𝑆 ↑ko 𝑅) ∈ Haus)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  {crab 3142   ∩ cin 3934   ⊆ wss 3935  ∅c0 4290  𝒫 cpw 4538  {csn 4560  ∪ cuni 4831  dom cdm 5549   “ cima 5552   Fn wfn 6344  ⟶wf 6345  ‘cfv 6349  (class class class)co 7150  Fincfn 8503   ↾_t crest 16688  Topctop 21495  TopOnctopon 21512   Cn ccn 21826  Hauscha 21910  Compccmp 21988   ↑_ko cxko 22163

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8283  df-map 8402  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-fi 8869  df-rest 16690  df-topgen 16711  df-top 21496  df-topon 21513  df-bases 21548  df-cn 21829  df-haus 21917  df-cmp 21989  df-xko 22165

This theorem is referenced by: (None)