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Theorem xkoccn 23506

Description: The "constant function" function which maps 𝑥 ∈ 𝑌 to the constant function 𝑧 ∈ 𝑋 ↦ 𝑥 is a continuous function from 𝑋 into the space of continuous functions from 𝑌 to 𝑋. This can also be understood as the currying of the first projection function. (The currying of the second projection function is 𝑥 ∈ 𝑌 ↦ (𝑧 ∈ 𝑋 ↦ 𝑧), which we already know is continuous because it is a constant function.) (Contributed by Mario Carneiro, 19-Mar-2015.)

**Assertion**
Ref	Expression
xkoccn	⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑥 ∈ 𝑌 ↦ (𝑋 × {𝑥})) ∈ (𝑆 Cn (𝑆 ↑_ko 𝑅)))

Distinct variable groups: 𝑥,𝑅 𝑥,𝑆 𝑥,𝑋 𝑥,𝑌

Proof of Theorem xkoccn Dummy variables 𝑓 𝑘 𝑣 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnconst2 23170	. . . 4 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌) ∧ 𝑥 ∈ 𝑌) → (𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆))
2	1	3expa 1118	. . 3 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝑌) → (𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆))
3	2	fmpttd 7087	. 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑥 ∈ 𝑌 ↦ (𝑋 × {𝑥})):𝑌⟶(𝑅 Cn 𝑆))
4		eqid 2729	. . . . . 6 ⊢ ∪ 𝑅 = ∪ 𝑅
5		eqid 2729	. . . . . 6 ⊢ {𝑧 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑧) ∈ Comp} = {𝑧 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑧) ∈ Comp}
6		eqid 2729	. . . . . 6 ⊢ (𝑘 ∈ {𝑧 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑧) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) = (𝑘 ∈ {𝑧 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑧) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})
7	4, 5, 6	xkobval 23473	. . . . 5 ⊢ ran (𝑘 ∈ {𝑧 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑧) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) = {𝑦 ∣ ∃𝑘 ∈ 𝒫 ∪ 𝑅∃𝑣 ∈ 𝑆 ((𝑅 ↾_t 𝑘) ∈ Comp ∧ 𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})}
8	7	eqabri 2871	. . . 4 ⊢ (𝑦 ∈ ran (𝑘 ∈ {𝑧 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑧) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ↔ ∃𝑘 ∈ 𝒫 ∪ 𝑅∃𝑣 ∈ 𝑆 ((𝑅 ↾_t 𝑘) ∈ Comp ∧ 𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))
9	2	ad5ant15 758	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) ∧ 𝑥 ∈ 𝑌) → (𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆))
10		simplr 768	. . . . . . . . . . . . . 14 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) ∧ 𝑥 ∈ 𝑌) → 𝑘 = ∅)
11	10	imaeq2d 6031	. . . . . . . . . . . . 13 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) ∧ 𝑥 ∈ 𝑌) → ((𝑋 × {𝑥}) “ 𝑘) = ((𝑋 × {𝑥}) “ ∅))
12		ima0 6048	. . . . . . . . . . . . . 14 ⊢ ((𝑋 × {𝑥}) “ ∅) = ∅
13		0ss 4363	. . . . . . . . . . . . . 14 ⊢ ∅ ⊆ 𝑣
14	12, 13	eqsstri 3993	. . . . . . . . . . . . 13 ⊢ ((𝑋 × {𝑥}) “ ∅) ⊆ 𝑣
15	11, 14	eqsstrdi 3991	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) ∧ 𝑥 ∈ 𝑌) → ((𝑋 × {𝑥}) “ 𝑘) ⊆ 𝑣)
16		imaeq1 6026	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑋 × {𝑥}) → (𝑓 “ 𝑘) = ((𝑋 × {𝑥}) “ 𝑘))
17	16	sseq1d 3978	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑋 × {𝑥}) → ((𝑓 “ 𝑘) ⊆ 𝑣 ↔ ((𝑋 × {𝑥}) “ 𝑘) ⊆ 𝑣))
18	17	elrab 3659	. . . . . . . . . . . 12 ⊢ ((𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} ↔ ((𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆) ∧ ((𝑋 × {𝑥}) “ 𝑘) ⊆ 𝑣))
19	9, 15, 18	sylanbrc 583	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) ∧ 𝑥 ∈ 𝑌) → (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})
20	19	ralrimiva 3125	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) → ∀𝑥 ∈ 𝑌 (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})
21		rabid2 3439	. . . . . . . . . 10 ⊢ (𝑌 = {𝑥 ∈ 𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}} ↔ ∀𝑥 ∈ 𝑌 (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})
22	20, 21	sylibr 234	. . . . . . . . 9 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) → 𝑌 = {𝑥 ∈ 𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}})
23		simpllr 775	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) → 𝑆 ∈ (TopOn‘𝑌))
24		toponmax 22813	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (TopOn‘𝑌) → 𝑌 ∈ 𝑆)
25	23, 24	syl 17	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) → 𝑌 ∈ 𝑆)
26	25	adantr 480	. . . . . . . . 9 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) → 𝑌 ∈ 𝑆)
27	22, 26	eqeltrrd 2829	. . . . . . . 8 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) → {𝑥 ∈ 𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}} ∈ 𝑆)
28		ifnefalse 4500	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ≠ ∅ → if(𝑘 = ∅, 𝑌, 𝑣) = 𝑣)
29	28	ad2antlr 727	. . . . . . . . . . . . . 14 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥 ∈ 𝑌) → if(𝑘 = ∅, 𝑌, 𝑣) = 𝑣)
30	29	eleq2d 2814	. . . . . . . . . . . . 13 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥 ∈ 𝑌) → (𝑥 ∈ if(𝑘 = ∅, 𝑌, 𝑣) ↔ 𝑥 ∈ 𝑣))
31		vex 3451	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ V
32	31	snss 4749	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑣 ↔ {𝑥} ⊆ 𝑣)
33	30, 32	bitrdi 287	. . . . . . . . . . . . . 14 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥 ∈ 𝑌) → (𝑥 ∈ if(𝑘 = ∅, 𝑌, 𝑣) ↔ {𝑥} ⊆ 𝑣))
34		df-ima 5651	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋 × {𝑥}) “ 𝑘) = ran ((𝑋 × {𝑥}) ↾ 𝑘)
35		simplrl 776	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) → 𝑘 ∈ 𝒫 ∪ 𝑅)
36	35	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥 ∈ 𝑌) → 𝑘 ∈ 𝒫 ∪ 𝑅)
37	36	elpwid 4572	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥 ∈ 𝑌) → 𝑘 ⊆ ∪ 𝑅)
38		toponuni 22801	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝑅)
39	38	ad5antr 734	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥 ∈ 𝑌) → 𝑋 = ∪ 𝑅)
40	37, 39	sseqtrrd 3984	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥 ∈ 𝑌) → 𝑘 ⊆ 𝑋)
41		xpssres 5989	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ⊆ 𝑋 → ((𝑋 × {𝑥}) ↾ 𝑘) = (𝑘 × {𝑥}))
42	40, 41	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥 ∈ 𝑌) → ((𝑋 × {𝑥}) ↾ 𝑘) = (𝑘 × {𝑥}))
43	42	rneqd 5902	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥 ∈ 𝑌) → ran ((𝑋 × {𝑥}) ↾ 𝑘) = ran (𝑘 × {𝑥}))
44	34, 43	eqtrid 2776	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥 ∈ 𝑌) → ((𝑋 × {𝑥}) “ 𝑘) = ran (𝑘 × {𝑥}))
45		rnxp 6143	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ≠ ∅ → ran (𝑘 × {𝑥}) = {𝑥})
46	45	ad2antlr 727	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥 ∈ 𝑌) → ran (𝑘 × {𝑥}) = {𝑥})
47	44, 46	eqtrd 2764	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥 ∈ 𝑌) → ((𝑋 × {𝑥}) “ 𝑘) = {𝑥})
48	47	sseq1d 3978	. . . . . . . . . . . . . 14 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥 ∈ 𝑌) → (((𝑋 × {𝑥}) “ 𝑘) ⊆ 𝑣 ↔ {𝑥} ⊆ 𝑣))
49	2	ad5ant15 758	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥 ∈ 𝑌) → (𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆))
50	49	biantrurd 532	. . . . . . . . . . . . . 14 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥 ∈ 𝑌) → (((𝑋 × {𝑥}) “ 𝑘) ⊆ 𝑣 ↔ ((𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆) ∧ ((𝑋 × {𝑥}) “ 𝑘) ⊆ 𝑣)))
51	33, 48, 50	3bitr2d 307	. . . . . . . . . . . . 13 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥 ∈ 𝑌) → (𝑥 ∈ if(𝑘 = ∅, 𝑌, 𝑣) ↔ ((𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆) ∧ ((𝑋 × {𝑥}) “ 𝑘) ⊆ 𝑣)))
52	30, 51	bitr3d 281	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥 ∈ 𝑌) → (𝑥 ∈ 𝑣 ↔ ((𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆) ∧ ((𝑋 × {𝑥}) “ 𝑘) ⊆ 𝑣)))
53	52, 18	bitr4di 289	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥 ∈ 𝑌) → (𝑥 ∈ 𝑣 ↔ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))
54	53	rabbi2dva 4189	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) → (𝑌 ∩ 𝑣) = {𝑥 ∈ 𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}})
55		simplrr 777	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) → 𝑣 ∈ 𝑆)
56		toponss 22814	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ (TopOn‘𝑌) ∧ 𝑣 ∈ 𝑆) → 𝑣 ⊆ 𝑌)
57	23, 55, 56	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) → 𝑣 ⊆ 𝑌)
58	57	adantr 480	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) → 𝑣 ⊆ 𝑌)
59		sseqin2 4186	. . . . . . . . . . 11 ⊢ (𝑣 ⊆ 𝑌 ↔ (𝑌 ∩ 𝑣) = 𝑣)
60	58, 59	sylib 218	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) → (𝑌 ∩ 𝑣) = 𝑣)
61	54, 60	eqtr3d 2766	. . . . . . . . 9 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) → {𝑥 ∈ 𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}} = 𝑣)
62	55	adantr 480	. . . . . . . . 9 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) → 𝑣 ∈ 𝑆)
63	61, 62	eqeltrd 2828	. . . . . . . 8 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) → {𝑥 ∈ 𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}} ∈ 𝑆)
64	27, 63	pm2.61dane 3012	. . . . . . 7 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) → {𝑥 ∈ 𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}} ∈ 𝑆)
65		imaeq2 6027	. . . . . . . . 9 ⊢ (𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} → (^◡(𝑥 ∈ 𝑌 ↦ (𝑋 × {𝑥})) “ 𝑦) = (^◡(𝑥 ∈ 𝑌 ↦ (𝑋 × {𝑥})) “ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))
66		eqid 2729	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑌 ↦ (𝑋 × {𝑥})) = (𝑥 ∈ 𝑌 ↦ (𝑋 × {𝑥}))
67	66	mptpreima 6211	. . . . . . . . 9 ⊢ (^◡(𝑥 ∈ 𝑌 ↦ (𝑋 × {𝑥})) “ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) = {𝑥 ∈ 𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}}
68	65, 67	eqtrdi 2780	. . . . . . . 8 ⊢ (𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} → (^◡(𝑥 ∈ 𝑌 ↦ (𝑋 × {𝑥})) “ 𝑦) = {𝑥 ∈ 𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}})
69	68	eleq1d 2813	. . . . . . 7 ⊢ (𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} → ((^◡(𝑥 ∈ 𝑌 ↦ (𝑋 × {𝑥})) “ 𝑦) ∈ 𝑆 ↔ {𝑥 ∈ 𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}} ∈ 𝑆))
70	64, 69	syl5ibrcom 247	. . . . . 6 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) → (𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} → (^◡(𝑥 ∈ 𝑌 ↦ (𝑋 × {𝑥})) “ 𝑦) ∈ 𝑆))
71	70	expimpd 453	. . . . 5 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) → (((𝑅 ↾_t 𝑘) ∈ Comp ∧ 𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) → (^◡(𝑥 ∈ 𝑌 ↦ (𝑋 × {𝑥})) “ 𝑦) ∈ 𝑆))
72	71	rexlimdvva 3194	. . . 4 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (∃𝑘 ∈ 𝒫 ∪ 𝑅∃𝑣 ∈ 𝑆 ((𝑅 ↾_t 𝑘) ∈ Comp ∧ 𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) → (^◡(𝑥 ∈ 𝑌 ↦ (𝑋 × {𝑥})) “ 𝑦) ∈ 𝑆))
73	8, 72	biimtrid 242	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑦 ∈ ran (𝑘 ∈ {𝑧 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑧) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) → (^◡(𝑥 ∈ 𝑌 ↦ (𝑋 × {𝑥})) “ 𝑦) ∈ 𝑆))
74	73	ralrimiv 3124	. 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ∀𝑦 ∈ ran (𝑘 ∈ {𝑧 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑧) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})(^◡(𝑥 ∈ 𝑌 ↦ (𝑋 × {𝑥})) “ 𝑦) ∈ 𝑆)
75		simpr 484	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑆 ∈ (TopOn‘𝑌))
76		ovex 7420	. . . . . 6 ⊢ (𝑅 Cn 𝑆) ∈ V
77	76	pwex 5335	. . . . 5 ⊢ 𝒫 (𝑅 Cn 𝑆) ∈ V
78	4, 5, 6	xkotf 23472	. . . . . 6 ⊢ (𝑘 ∈ {𝑧 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑧) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}):({𝑧 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑧) ∈ Comp} × 𝑆)⟶𝒫 (𝑅 Cn 𝑆)
79		frn 6695	. . . . . 6 ⊢ ((𝑘 ∈ {𝑧 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑧) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}):({𝑧 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑧) ∈ Comp} × 𝑆)⟶𝒫 (𝑅 Cn 𝑆) → ran (𝑘 ∈ {𝑧 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑧) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ⊆ 𝒫 (𝑅 Cn 𝑆))
80	78, 79	ax-mp 5	. . . . 5 ⊢ ran (𝑘 ∈ {𝑧 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑧) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ⊆ 𝒫 (𝑅 Cn 𝑆)
81	77, 80	ssexi 5277	. . . 4 ⊢ ran (𝑘 ∈ {𝑧 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑧) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ∈ V
82	81	a1i 11	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ran (𝑘 ∈ {𝑧 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑧) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ∈ V)
83		topontop 22800	. . . 4 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑅 ∈ Top)
84		topontop 22800	. . . 4 ⊢ (𝑆 ∈ (TopOn‘𝑌) → 𝑆 ∈ Top)
85	4, 5, 6	xkoval 23474	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑_ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑧 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑧) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))))
86	83, 84, 85	syl2an 596	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑆 ↑_ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑧 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑧) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))))
87		eqid 2729	. . . . 5 ⊢ (𝑆 ↑_ko 𝑅) = (𝑆 ↑_ko 𝑅)
88	87	xkotopon 23487	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑_ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
89	83, 84, 88	syl2an 596	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑆 ↑_ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
90	75, 82, 86, 89	subbascn 23141	. 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ((𝑥 ∈ 𝑌 ↦ (𝑋 × {𝑥})) ∈ (𝑆 Cn (𝑆 ↑_ko 𝑅)) ↔ ((𝑥 ∈ 𝑌 ↦ (𝑋 × {𝑥})):𝑌⟶(𝑅 Cn 𝑆) ∧ ∀𝑦 ∈ ran (𝑘 ∈ {𝑧 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑧) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})(^◡(𝑥 ∈ 𝑌 ↦ (𝑋 × {𝑥})) “ 𝑦) ∈ 𝑆)))
91	3, 74, 90	mpbir2and 713	1 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑥 ∈ 𝑌 ↦ (𝑋 × {𝑥})) ∈ (𝑆 Cn (𝑆 ↑_ko 𝑅)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 {crab 3405 Vcvv 3447 ∩ cin 3913 ⊆ wss 3914 ∅c0 4296 ifcif 4488 𝒫 cpw 4563 {csn 4589 ∪ cuni 4871 ↦ cmpt 5188 × cxp 5636 ^◡ccnv 5637 ran crn 5639 ↾ cres 5640 “ cima 5641 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 ∈ cmpo 7389 ficfi 9361 ↾_t crest 17383 topGenctg 17400 Topctop 22780 TopOnctopon 22797 Cn ccn 23111 Compccmp 23273 ↑_ko cxko 23448

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-1o 8434 df-2o 8435 df-map 8801 df-en 8919 df-dom 8920 df-fin 8922 df-fi 9362 df-rest 17385 df-topgen 17406 df-top 22781 df-topon 22798 df-bases 22833 df-cn 23114 df-cnp 23115 df-cmp 23274 df-xko 23450

This theorem is referenced by: cnmptkc 23566 xkofvcn 23571

W3C validator