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Theorem xkoccn 23614

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 23278	. . . 4 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌) ∧ 𝑥 ∈ 𝑌) → (𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆))
2	1	3expa 1115	. . 3 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝑌) → (𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆))
3	2	fmpttd 7129	. 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑥 ∈ 𝑌 ↦ (𝑋 × {𝑥})):𝑌⟶(𝑅 Cn 𝑆))
4		eqid 2726	. . . . . 6 ⊢ ∪ 𝑅 = ∪ 𝑅
5		eqid 2726	. . . . . 6 ⊢ {𝑧 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑧) ∈ Comp} = {𝑧 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾_t 𝑧) ∈ Comp}
6		eqid 2726	. . . . . 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 23581	. . . . 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 2870	. . . 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 757	. . . . . . . . . . . 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 767	. . . . . . . . . . . . . 14 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) ∧ 𝑥 ∈ 𝑌) → 𝑘 = ∅)
11	10	imaeq2d 6069	. . . . . . . . . . . . 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 6086	. . . . . . . . . . . . . 14 ⊢ ((𝑋 × {𝑥}) “ ∅) = ∅
13		0ss 4401	. . . . . . . . . . . . . 14 ⊢ ∅ ⊆ 𝑣
14	12, 13	eqsstri 4014	. . . . . . . . . . . . 13 ⊢ ((𝑋 × {𝑥}) “ ∅) ⊆ 𝑣
15	11, 14	eqsstrdi 4034	. . . . . . . . . . . 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 6064	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑋 × {𝑥}) → (𝑓 “ 𝑘) = ((𝑋 × {𝑥}) “ 𝑘))
17	16	sseq1d 4011	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑋 × {𝑥}) → ((𝑓 “ 𝑘) ⊆ 𝑣 ↔ ((𝑋 × {𝑥}) “ 𝑘) ⊆ 𝑣))
18	17	elrab 3681	. . . . . . . . . . . 12 ⊢ ((𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} ↔ ((𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆) ∧ ((𝑋 × {𝑥}) “ 𝑘) ⊆ 𝑣))
19	9, 15, 18	sylanbrc 581	. . . . . . . . . . 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 3136	. . . . . . . . . 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 3453	. . . . . . . . . 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 774	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) → 𝑆 ∈ (TopOn‘𝑌))
24		toponmax 22919	. . . . . . . . . . 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 479	. . . . . . . . 9 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) → 𝑌 ∈ 𝑆)
27	22, 26	eqeltrrd 2827	. . . . . . . 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 4545	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ≠ ∅ → if(𝑘 = ∅, 𝑌, 𝑣) = 𝑣)
29	28	ad2antlr 725	. . . . . . . . . . . . . 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 2812	. . . . . . . . . . . . 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 3466	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ V
32	31	snss 4794	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑣 ↔ {𝑥} ⊆ 𝑣)
33	30, 32	bitrdi 286	. . . . . . . . . . . . . 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 5695	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋 × {𝑥}) “ 𝑘) = ran ((𝑋 × {𝑥}) ↾ 𝑘)
35		simplrl 775	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) → 𝑘 ∈ 𝒫 ∪ 𝑅)
36	35	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . 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 4616	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥 ∈ 𝑌) → 𝑘 ⊆ ∪ 𝑅)
38		toponuni 22907	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝑅)
39	38	ad5antr 732	. . . . . . . . . . . . . . . . . . . 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 4021	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥 ∈ 𝑌) → 𝑘 ⊆ 𝑋)
41		xpssres 6027	. . . . . . . . . . . . . . . . . . 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 5944	. . . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . . 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 6181	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ≠ ∅ → ran (𝑘 × {𝑥}) = {𝑥})
46	45	ad2antlr 725	. . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . 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 4011	. . . . . . . . . . . . . 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 757	. . . . . . . . . . . . . . 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 531	. . . . . . . . . . . . . 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 306	. . . . . . . . . . . . 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 280	. . . . . . . . . . . 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 288	. . . . . . . . . . 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 4219	. . . . . . . . . 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 776	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) → 𝑣 ∈ 𝑆)
56		toponss 22920	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ (TopOn‘𝑌) ∧ 𝑣 ∈ 𝑆) → 𝑣 ⊆ 𝑌)
57	23, 55, 56	syl2anc 582	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) → 𝑣 ⊆ 𝑌)
58	57	adantr 479	. . . . . . . . . . 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 2768	. . . . . . . . 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 479	. . . . . . . . 9 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾_t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) → 𝑣 ∈ 𝑆)
63	61, 62	eqeltrd 2826	. . . . . . . 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 3019	. . . . . . 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 6065	. . . . . . . . 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 2726	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑌 ↦ (𝑋 × {𝑥})) = (𝑥 ∈ 𝑌 ↦ (𝑋 × {𝑥}))
67	66	mptpreima 6249	. . . . . . . . 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 2782	. . . . . . . 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 2811	. . . . . . 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 452	. . . . 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 3202	. . . 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 3135	. 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 483	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑆 ∈ (TopOn‘𝑌))
76		ovex 7457	. . . . . 6 ⊢ (𝑅 Cn 𝑆) ∈ V
77	76	pwex 5384	. . . . 5 ⊢ 𝒫 (𝑅 Cn 𝑆) ∈ V
78	4, 5, 6	xkotf 23580	. . . . . 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 6735	. . . . . 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 5327	. . . 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 22906	. . . 4 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑅 ∈ Top)
84		topontop 22906	. . . 4 ⊢ (𝑆 ∈ (TopOn‘𝑌) → 𝑆 ∈ Top)
85	4, 5, 6	xkoval 23582	. . . 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 594	. . 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 2726	. . . . 5 ⊢ (𝑆 ↑_ko 𝑅) = (𝑆 ↑_ko 𝑅)
88	87	xkotopon 23595	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑_ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
89	83, 84, 88	syl2an 594	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑆 ↑_ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
90	75, 82, 86, 89	subbascn 23249	. 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 711	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 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∀wral 3051 ∃wrex 3060 {crab 3419 Vcvv 3462 ∩ cin 3946 ⊆ wss 3947 ∅c0 4325 ifcif 4533 𝒫 cpw 4607 {csn 4633 ∪ cuni 4913 ↦ cmpt 5236 × cxp 5680 ^◡ccnv 5681 ran crn 5683 ↾ cres 5684 “ cima 5685 ⟶wf 6550 ‘cfv 6554 (class class class)co 7424 ∈ cmpo 7426 ficfi 9453 ↾_t crest 17435 topGenctg 17452 Topctop 22886 TopOnctopon 22903 Cn ccn 23219 Compccmp 23381 ↑_ko cxko 23556

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-1o 8496 df-2o 8497 df-map 8857 df-en 8975 df-dom 8976 df-fin 8978 df-fi 9454 df-rest 17437 df-topgen 17458 df-top 22887 df-topon 22904 df-bases 22940 df-cn 23222 df-cnp 23223 df-cmp 23382 df-xko 23558

This theorem is referenced by: cnmptkc 23674 xkofvcn 23679

W3C validator