Theorem xkoccn 23745

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 23409	. . . 4 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌) ∧ 𝑥 ∈ 𝑌) → (𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆))
2	1	3expa 1134	. . 3 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝑌) → (𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆))
3	2	fmpttd 7111	. 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑥 ∈ 𝑌 ↦ (𝑋 × {𝑥})):𝑌⟶(𝑅 Cn 𝑆))
4		eqid 2769	. . . . . 6 ⊢ ∪ 𝑅 = ∪ 𝑅
5		eqid 2769	. . . . . 6 ⊢ {𝑧 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑧) ∈ Comp} = {𝑧 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑧) ∈ Comp}
6		eqid 2769	. . . . . 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 23712	. . . . 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 2911	. . . 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 770	. . . . . . . . . . . 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 780	. . . . . . . . . . . . . 14 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) ∧ 𝑥 ∈ 𝑌) → 𝑘 = ∅)
11	10	imaeq2d 6063	. . . . . . . . . . . . 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 6080	. . . . . . . . . . . . . 14 ⊢ ((𝑋 × {𝑥}) “ ∅) = ∅
13		0ss 4364	. . . . . . . . . . . . . 14 ⊢ ∅ ⊆ 𝑣
14	12, 13	eqsstri 3991	. . . . . . . . . . . . 13 ⊢ ((𝑋 × {𝑥}) “ ∅) ⊆ 𝑣
15	11, 14	eqsstrdi 3989	. . . . . . . . . . . 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 6058	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑋 × {𝑥}) → (𝑓 “ 𝑘) = ((𝑋 × {𝑥}) “ 𝑘))
17	16	sseq1d 3976	. . . . . . . . . . . . 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 594	. . . . . . . . . . 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 3163	. . . . . . . . . 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 3456	. . . . . . . . . 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 237	. . . . . . . . 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 787	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → 𝑆 ∈ (TopOn‘𝑌))
24		toponmax 23052	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (TopOn‘𝑌) → 𝑌 ∈ 𝑆)
25	23, 24	syl 18	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → 𝑌 ∈ 𝑆)
26	25	adantr 485	. . . . . . . . 9 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) → 𝑌 ∈ 𝑆)
27	22, 26	eqeltrrd 2870	. . . . . . . 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 4504	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ≠ ∅ → if(𝑘 = ∅, 𝑌, 𝑣) = 𝑣)
29	28	ad2antlr 739	. . . . . . . . . . . . . 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 2855	. . . . . . . . . . . . 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 3467	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ V
32	31	snss 4755	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑣 ↔ {𝑥} ⊆ 𝑣)
33	30, 32	bitrdi 290	. . . . . . . . . . . . . 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 5675	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋 × {𝑥}) “ 𝑘) = ran ((𝑋 × {𝑥}) ↾ 𝑘)
35		simplrl 788	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → 𝑘 ∈ 𝒫 ∪ 𝑅)
36	35	ad2antrr 738	. . . . . . . . . . . . . . . . . . . . 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 4576	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥 ∈ 𝑌) → 𝑘 ⊆ ∪ 𝑅)
38		toponuni 23040	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝑅)
39	38	ad5antr 746	. . . . . . . . . . . . . . . . . . . 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 3982	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥 ∈ 𝑌) → 𝑘 ⊆ 𝑋)
41		xpssres 6018	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ⊆ 𝑋 → ((𝑋 × {𝑥}) ↾ 𝑘) = (𝑘 × {𝑥}))
42	40, 41	syl 18	. . . . . . . . . . . . . . . . . 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 5929	. . . . . . . . . . . . . . . . 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 2816	. . . . . . . . . . . . . . . 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 6169	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ≠ ∅ → ran (𝑘 × {𝑥}) = {𝑥})
46	45	ad2antlr 739	. . . . . . . . . . . . . . . 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 2804	. . . . . . . . . . . . . . 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 3976	. . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . 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 541	. . . . . . . . . . . . . 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 310	. . . . . . . . . . . . 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 284	. . . . . . . . . . . 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 292	. . . . . . . . . . 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 4186	. . . . . . . . . 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 789	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → 𝑣 ∈ 𝑆)
56		toponss 23053	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ (TopOn‘𝑌) ∧ 𝑣 ∈ 𝑆) → 𝑣 ⊆ 𝑌)
57	23, 55, 56	syl2anc 595	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) → 𝑣 ⊆ 𝑌)
58	57	adantr 485	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) → 𝑣 ⊆ 𝑌)
59		sseqin2 4184	. . . . . . . . . . 11 ⊢ (𝑣 ⊆ 𝑌 ↔ (𝑌 ∩ 𝑣) = 𝑣)
60	58, 59	sylib 221	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) → (𝑌 ∩ 𝑣) = 𝑣)
61	54, 60	eqtr3d 2806	. . . . . . . . 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 485	. . . . . . . . 9 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑅 ∧ 𝑣 ∈ 𝑆)) ∧ (𝑅 ↾t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) → 𝑣 ∈ 𝑆)
63	61, 62	eqeltrd 2869	. . . . . . . 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 3051	. . . . . . 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 6059	. . . . . . . . 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 2769	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑌 ↦ (𝑋 × {𝑥})) = (𝑥 ∈ 𝑌 ↦ (𝑋 × {𝑥}))
67	66	mptpreima 6240	. . . . . . . . 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 2820	. . . . . . . 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 2854	. . . . . . 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 250	. . . . . 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 458	. . . . 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 3228	. . . 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 245	. . 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 3162	. 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 489	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑆 ∈ (TopOn‘𝑌))
76		ovex 7444	. . . . . 6 ⊢ (𝑅 Cn 𝑆) ∈ V
77	76	pwex 5352	. . . . 5 ⊢ 𝒫 (𝑅 Cn 𝑆) ∈ V
78	4, 5, 6	xkotf 23711	. . . . . 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 6714	. . . . . 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 5293	. . . 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 23039	. . . 4 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑅 ∈ Top)
84		topontop 23039	. . . 4 ⊢ (𝑆 ∈ (TopOn‘𝑌) → 𝑆 ∈ Top)
85	4, 5, 6	xkoval 23713	. . . 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 607	. . 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 2769	. . . . 5 ⊢ (𝑆 ↑ko 𝑅) = (𝑆 ↑ko 𝑅)
88	87	xkotopon 23726	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
89	83, 84, 88	syl2an 607	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑆 ↑ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
90	75, 82, 86, 89	subbascn 23380	. 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 725	1 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑥 ∈ 𝑌 ↦ (𝑋 × {𝑥})) ∈ (𝑆 Cn (𝑆 ↑ko 𝑅)))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095  {crab 3423  Vcvv 3463   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294  ifcif 4492  𝒫 cpw 4567  {csn 4594  ∪ cuni 4876   ↦ cmpt 5196   × cxp 5660  ^◡ccnv 5661  ran crn 5663   ↾ cres 5664   “ cima 5665  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413  ficfi 9370   ↾_t crest 17473  topGenctg 17490  Topctop 23019  TopOnctopon 23036   Cn ccn 23350  Compccmp 23512   ↑_ko cxko 23687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-1o 8453  df-2o 8454  df-map 8826  df-en 8944  df-dom 8945  df-fin 8947  df-fi 9371  df-rest 17475  df-topgen 17496  df-top 23020  df-topon 23037  df-bases 23072  df-cn 23353  df-cnp 23354  df-cmp 23513  df-xko 23689

This theorem is referenced by:  cnmptkc  23805  xkofvcn  23810