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Theorem xkoccn 23123

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 22787	. . . 4 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌) ∧ 𝑥 ∈ 𝑌) → (𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆))
2	1	3expa 1119	. . 3 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝑌) → (𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆))
3	2	fmpttd 7115	. 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑥 ∈ 𝑌 ↦ (𝑋 × {𝑥})):𝑌⟶(𝑅 Cn 𝑆))
4		eqid 2733	. . . . . 6 ⊢ ∪ 𝑅 = ∪ 𝑅
5		eqid 2733	. . . . . 6 ⊢ {𝑧 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑧) ∈ Comp} = {𝑧 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑧) ∈ Comp}
6		eqid 2733	. . . . . 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 23090	. . . . 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 2878	. . . 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 6060	. . . . . . . . . . . . 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 6077	. . . . . . . . . . . . . 14 ⊢ ((𝑋 × {𝑥}) “ ∅) = ∅
13		0ss 4397	. . . . . . . . . . . . . 14 ⊢ ∅ ⊆ 𝑣
14	12, 13	eqsstri 4017	. . . . . . . . . . . . 13 ⊢ ((𝑋 × {𝑥}) “ ∅) ⊆ 𝑣
15	11, 14	eqsstrdi 4037	. . . . . . . . . . . 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 6055	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑋 × {𝑥}) → (𝑓 “ 𝑘) = ((𝑋 × {𝑥}) “ 𝑘))
17	16	sseq1d 4014	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑋 × {𝑥}) → ((𝑓 “ 𝑘) ⊆ 𝑣 ↔ ((𝑋 × {𝑥}) “ 𝑘) ⊆ 𝑣))
18	17	elrab 3684	. . . . . . . . . . . 12 ⊢ ((𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} ↔ ((𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆) ∧ ((𝑋 × {𝑥}) “ 𝑘) ⊆ 𝑣))
19	9, 15, 18	sylanbrc 584	. . . . . . . . . . 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 3147	. . . . . . . . . 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 3465	. . . . . . . . . 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 233	. . . . . . . . 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 22428	. . . . . . . . . . 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 482	. . . . . . . . 9 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) → 𝑌 ∈ 𝑆)
27	22, 26	eqeltrrd 2835	. . . . . . . 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 4541	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ≠ ∅ → if(𝑘 = ∅, 𝑌, 𝑣) = 𝑣)
29	28	ad2antlr 726	. . . . . . . . . . . . . 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 2820	. . . . . . . . . . . . 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 3479	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ V
32	31	snss 4790	. . . . . . . . . . . . . . 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 5690	. . . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . . . . . . . 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 4612	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥 ∈ 𝑌) → 𝑘 ⊆ ∪ 𝑅)
38		toponuni 22416	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝑅)
39	38	ad5antr 733	. . . . . . . . . . . . . . . . . . . 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 4024	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥 ∈ 𝑌) → 𝑘 ⊆ 𝑋)
41		xpssres 6019	. . . . . . . . . . . . . . . . . . 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 5938	. . . . . . . . . . . . . . . . 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 2785	. . . . . . . . . . . . . . . 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 6170	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ≠ ∅ → ran (𝑘 × {𝑥}) = {𝑥})
46	45	ad2antlr 726	. . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . 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 4014	. . . . . . . . . . . . . 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 534	. . . . . . . . . . . . . 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 4218	. . . . . . . . . 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 22429	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ (TopOn‘𝑌) ∧ 𝑣 ∈ 𝑆) → 𝑣 ⊆ 𝑌)
57	23, 55, 56	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) → 𝑣 ⊆ 𝑌)
58	57	adantr 482	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) → 𝑣 ⊆ 𝑌)
59		sseqin2 4216	. . . . . . . . . . 11 ⊢ (𝑣 ⊆ 𝑌 ↔ (𝑌 ∩ 𝑣) = 𝑣)
60	58, 59	sylib 217	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) → (𝑌 ∩ 𝑣) = 𝑣)
61	54, 60	eqtr3d 2775	. . . . . . . . 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 482	. . . . . . . . 9 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) → 𝑣 ∈ 𝑆)
63	61, 62	eqeltrd 2834	. . . . . . . 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 3030	. . . . . . 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 6056	. . . . . . . . 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 2733	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑌 ↦ (𝑋 × {𝑥})) = (𝑥 ∈ 𝑌 ↦ (𝑋 × {𝑥}))
67	66	mptpreima 6238	. . . . . . . . 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 2789	. . . . . . . 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 2819	. . . . . . 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 246	. . . . . 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 455	. . . . 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 3212	. . . 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 241	. . 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 3146	. 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 486	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑆 ∈ (TopOn‘𝑌))
76		ovex 7442	. . . . . 6 ⊢ (𝑅 Cn 𝑆) ∈ V
77	76	pwex 5379	. . . . 5 ⊢ 𝒫 (𝑅 Cn 𝑆) ∈ V
78	4, 5, 6	xkotf 23089	. . . . . 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 6725	. . . . . 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 5323	. . . 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 22415	. . . 4 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑅 ∈ Top)
84		topontop 22415	. . . 4 ⊢ (𝑆 ∈ (TopOn‘𝑌) → 𝑆 ∈ Top)
85	4, 5, 6	xkoval 23091	. . . 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 597	. . 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 2733	. . . . 5 ⊢ (𝑆 ↑_ko 𝑅) = (𝑆 ↑_ko 𝑅)
88	87	xkotopon 23104	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑_ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
89	83, 84, 88	syl2an 597	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑆 ↑_ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
90	75, 82, 86, 89	subbascn 22758	. 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 712	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 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 {crab 3433 Vcvv 3475 ∩ cin 3948 ⊆ wss 3949 ∅c0 4323 ifcif 4529 𝒫 cpw 4603 {csn 4629 ∪ cuni 4909 ↦ cmpt 5232 × cxp 5675 ^◡ccnv 5676 ran crn 5678 ↾ cres 5679 “ cima 5680 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ∈ cmpo 7411 ficfi 9405 ↾_t crest 17366 topGenctg 17383 Topctop 22395 TopOnctopon 22412 Cn ccn 22728 Compccmp 22890 ↑_ko cxko 23065

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-1o 8466 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-fin 8943 df-fi 9406 df-rest 17368 df-topgen 17389 df-top 22396 df-topon 22413 df-bases 22449 df-cn 22731 df-cnp 22732 df-cmp 22891 df-xko 23067

This theorem is referenced by: cnmptkc 23183 xkofvcn 23188

W3C validator