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Theorem xkoccn 22970
Description: The "constant function" function which maps 𝑥𝑌 to the constant function 𝑧𝑋𝑥 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 𝑥𝑌 ↦ (𝑧𝑋𝑧), 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)) → (𝑥𝑌 ↦ (𝑋 × {𝑥})) ∈ (𝑆 Cn (𝑆ko 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.
StepHypRef Expression
1 cnconst2 22634 . . . 4 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌) ∧ 𝑥𝑌) → (𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆))
213expa 1118 . . 3 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑥𝑌) → (𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆))
32fmpttd 7063 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑥𝑌 ↦ (𝑋 × {𝑥})):𝑌⟶(𝑅 Cn 𝑆))
4 eqid 2736 . . . . . 6 𝑅 = 𝑅
5 eqid 2736 . . . . . 6 {𝑧 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑧) ∈ Comp} = {𝑧 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑧) ∈ Comp}
6 eqid 2736 . . . . . 6 (𝑘 ∈ {𝑧 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑧) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) = (𝑘 ∈ {𝑧 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑧) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})
74, 5, 6xkobval 22937 . . . . 5 ran (𝑘 ∈ {𝑧 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑧) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) = {𝑦 ∣ ∃𝑘 ∈ 𝒫 𝑅𝑣𝑆 ((𝑅t 𝑘) ∈ Comp ∧ 𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})}
87eqabi 2881 . . . 4 (𝑦 ∈ ran (𝑘 ∈ {𝑧 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑧) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) ↔ ∃𝑘 ∈ 𝒫 𝑅𝑣𝑆 ((𝑅t 𝑘) ∈ Comp ∧ 𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))
92ad5ant15 757 . . . . . . . . . . . 12 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) ∧ 𝑥𝑌) → (𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆))
10 simplr 767 . . . . . . . . . . . . . 14 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) ∧ 𝑥𝑌) → 𝑘 = ∅)
1110imaeq2d 6013 . . . . . . . . . . . . 13 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) ∧ 𝑥𝑌) → ((𝑋 × {𝑥}) “ 𝑘) = ((𝑋 × {𝑥}) “ ∅))
12 ima0 6029 . . . . . . . . . . . . . 14 ((𝑋 × {𝑥}) “ ∅) = ∅
13 0ss 4356 . . . . . . . . . . . . . 14 ∅ ⊆ 𝑣
1412, 13eqsstri 3978 . . . . . . . . . . . . 13 ((𝑋 × {𝑥}) “ ∅) ⊆ 𝑣
1511, 14eqsstrdi 3998 . . . . . . . . . . . 12 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) ∧ 𝑥𝑌) → ((𝑋 × {𝑥}) “ 𝑘) ⊆ 𝑣)
16 imaeq1 6008 . . . . . . . . . . . . . 14 (𝑓 = (𝑋 × {𝑥}) → (𝑓𝑘) = ((𝑋 × {𝑥}) “ 𝑘))
1716sseq1d 3975 . . . . . . . . . . . . 13 (𝑓 = (𝑋 × {𝑥}) → ((𝑓𝑘) ⊆ 𝑣 ↔ ((𝑋 × {𝑥}) “ 𝑘) ⊆ 𝑣))
1817elrab 3645 . . . . . . . . . . . 12 ((𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} ↔ ((𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆) ∧ ((𝑋 × {𝑥}) “ 𝑘) ⊆ 𝑣))
199, 15, 18sylanbrc 583 . . . . . . . . . . 11 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) ∧ 𝑥𝑌) → (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})
2019ralrimiva 3143 . . . . . . . . . 10 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) → ∀𝑥𝑌 (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})
21 rabid2 3436 . . . . . . . . . 10 (𝑌 = {𝑥𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}} ↔ ∀𝑥𝑌 (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})
2220, 21sylibr 233 . . . . . . . . 9 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) → 𝑌 = {𝑥𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}})
23 simpllr 774 . . . . . . . . . . 11 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑆 ∈ (TopOn‘𝑌))
24 toponmax 22275 . . . . . . . . . . 11 (𝑆 ∈ (TopOn‘𝑌) → 𝑌𝑆)
2523, 24syl 17 . . . . . . . . . 10 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑌𝑆)
2625adantr 481 . . . . . . . . 9 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) → 𝑌𝑆)
2722, 26eqeltrrd 2839 . . . . . . . 8 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) → {𝑥𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}} ∈ 𝑆)
28 ifnefalse 4498 . . . . . . . . . . . . . . 15 (𝑘 ≠ ∅ → if(𝑘 = ∅, 𝑌, 𝑣) = 𝑣)
2928ad2antlr 725 . . . . . . . . . . . . . 14 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → if(𝑘 = ∅, 𝑌, 𝑣) = 𝑣)
3029eleq2d 2823 . . . . . . . . . . . . 13 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → (𝑥 ∈ if(𝑘 = ∅, 𝑌, 𝑣) ↔ 𝑥𝑣))
31 vex 3449 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
3231snss 4746 . . . . . . . . . . . . . . 15 (𝑥𝑣 ↔ {𝑥} ⊆ 𝑣)
3330, 32bitrdi 286 . . . . . . . . . . . . . 14 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → (𝑥 ∈ if(𝑘 = ∅, 𝑌, 𝑣) ↔ {𝑥} ⊆ 𝑣))
34 df-ima 5646 . . . . . . . . . . . . . . . . 17 ((𝑋 × {𝑥}) “ 𝑘) = ran ((𝑋 × {𝑥}) ↾ 𝑘)
35 simplrl 775 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑘 ∈ 𝒫 𝑅)
3635ad2antrr 724 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → 𝑘 ∈ 𝒫 𝑅)
3736elpwid 4569 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → 𝑘 𝑅)
38 toponuni 22263 . . . . . . . . . . . . . . . . . . . . 21 (𝑅 ∈ (TopOn‘𝑋) → 𝑋 = 𝑅)
3938ad5antr 732 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → 𝑋 = 𝑅)
4037, 39sseqtrrd 3985 . . . . . . . . . . . . . . . . . . 19 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → 𝑘𝑋)
41 xpssres 5974 . . . . . . . . . . . . . . . . . . 19 (𝑘𝑋 → ((𝑋 × {𝑥}) ↾ 𝑘) = (𝑘 × {𝑥}))
4240, 41syl 17 . . . . . . . . . . . . . . . . . 18 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → ((𝑋 × {𝑥}) ↾ 𝑘) = (𝑘 × {𝑥}))
4342rneqd 5893 . . . . . . . . . . . . . . . . 17 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → ran ((𝑋 × {𝑥}) ↾ 𝑘) = ran (𝑘 × {𝑥}))
4434, 43eqtrid 2788 . . . . . . . . . . . . . . . 16 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → ((𝑋 × {𝑥}) “ 𝑘) = ran (𝑘 × {𝑥}))
45 rnxp 6122 . . . . . . . . . . . . . . . . 17 (𝑘 ≠ ∅ → ran (𝑘 × {𝑥}) = {𝑥})
4645ad2antlr 725 . . . . . . . . . . . . . . . 16 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → ran (𝑘 × {𝑥}) = {𝑥})
4744, 46eqtrd 2776 . . . . . . . . . . . . . . 15 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → ((𝑋 × {𝑥}) “ 𝑘) = {𝑥})
4847sseq1d 3975 . . . . . . . . . . . . . 14 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → (((𝑋 × {𝑥}) “ 𝑘) ⊆ 𝑣 ↔ {𝑥} ⊆ 𝑣))
492ad5ant15 757 . . . . . . . . . . . . . . 15 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → (𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆))
5049biantrurd 533 . . . . . . . . . . . . . 14 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → (((𝑋 × {𝑥}) “ 𝑘) ⊆ 𝑣 ↔ ((𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆) ∧ ((𝑋 × {𝑥}) “ 𝑘) ⊆ 𝑣)))
5133, 48, 503bitr2d 306 . . . . . . . . . . . . 13 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → (𝑥 ∈ if(𝑘 = ∅, 𝑌, 𝑣) ↔ ((𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆) ∧ ((𝑋 × {𝑥}) “ 𝑘) ⊆ 𝑣)))
5230, 51bitr3d 280 . . . . . . . . . . . 12 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → (𝑥𝑣 ↔ ((𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆) ∧ ((𝑋 × {𝑥}) “ 𝑘) ⊆ 𝑣)))
5352, 18bitr4di 288 . . . . . . . . . . 11 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → (𝑥𝑣 ↔ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))
5453rabbi2dva 4177 . . . . . . . . . 10 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) → (𝑌𝑣) = {𝑥𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}})
55 simplrr 776 . . . . . . . . . . . . 13 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑣𝑆)
56 toponss 22276 . . . . . . . . . . . . 13 ((𝑆 ∈ (TopOn‘𝑌) ∧ 𝑣𝑆) → 𝑣𝑌)
5723, 55, 56syl2anc 584 . . . . . . . . . . . 12 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑣𝑌)
5857adantr 481 . . . . . . . . . . 11 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) → 𝑣𝑌)
59 sseqin2 4175 . . . . . . . . . . 11 (𝑣𝑌 ↔ (𝑌𝑣) = 𝑣)
6058, 59sylib 217 . . . . . . . . . 10 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) → (𝑌𝑣) = 𝑣)
6154, 60eqtr3d 2778 . . . . . . . . 9 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) → {𝑥𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}} = 𝑣)
6255adantr 481 . . . . . . . . 9 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) → 𝑣𝑆)
6361, 62eqeltrd 2838 . . . . . . . 8 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) → {𝑥𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}} ∈ 𝑆)
6427, 63pm2.61dane 3032 . . . . . . 7 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) → {𝑥𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}} ∈ 𝑆)
65 imaeq2 6009 . . . . . . . . 9 (𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} → ((𝑥𝑌 ↦ (𝑋 × {𝑥})) “ 𝑦) = ((𝑥𝑌 ↦ (𝑋 × {𝑥})) “ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))
66 eqid 2736 . . . . . . . . . 10 (𝑥𝑌 ↦ (𝑋 × {𝑥})) = (𝑥𝑌 ↦ (𝑋 × {𝑥}))
6766mptpreima 6190 . . . . . . . . 9 ((𝑥𝑌 ↦ (𝑋 × {𝑥})) “ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) = {𝑥𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}}
6865, 67eqtrdi 2792 . . . . . . . 8 (𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} → ((𝑥𝑌 ↦ (𝑋 × {𝑥})) “ 𝑦) = {𝑥𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}})
6968eleq1d 2822 . . . . . . 7 (𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} → (((𝑥𝑌 ↦ (𝑋 × {𝑥})) “ 𝑦) ∈ 𝑆 ↔ {𝑥𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}} ∈ 𝑆))
7064, 69syl5ibrcom 246 . . . . . 6 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) → (𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} → ((𝑥𝑌 ↦ (𝑋 × {𝑥})) “ 𝑦) ∈ 𝑆))
7170expimpd 454 . . . . 5 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) → (((𝑅t 𝑘) ∈ Comp ∧ 𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) → ((𝑥𝑌 ↦ (𝑋 × {𝑥})) “ 𝑦) ∈ 𝑆))
7271rexlimdvva 3205 . . . 4 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (∃𝑘 ∈ 𝒫 𝑅𝑣𝑆 ((𝑅t 𝑘) ∈ Comp ∧ 𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) → ((𝑥𝑌 ↦ (𝑋 × {𝑥})) “ 𝑦) ∈ 𝑆))
738, 72biimtrid 241 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑦 ∈ ran (𝑘 ∈ {𝑧 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑧) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) → ((𝑥𝑌 ↦ (𝑋 × {𝑥})) “ 𝑦) ∈ 𝑆))
7473ralrimiv 3142 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ∀𝑦 ∈ ran (𝑘 ∈ {𝑧 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑧) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})((𝑥𝑌 ↦ (𝑋 × {𝑥})) “ 𝑦) ∈ 𝑆)
75 simpr 485 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑆 ∈ (TopOn‘𝑌))
76 ovex 7390 . . . . . 6 (𝑅 Cn 𝑆) ∈ V
7776pwex 5335 . . . . 5 𝒫 (𝑅 Cn 𝑆) ∈ V
784, 5, 6xkotf 22936 . . . . . 6 (𝑘 ∈ {𝑧 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑧) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}):({𝑧 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑧) ∈ Comp} × 𝑆)⟶𝒫 (𝑅 Cn 𝑆)
79 frn 6675 . . . . . 6 ((𝑘 ∈ {𝑧 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑧) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}):({𝑧 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑧) ∈ Comp} × 𝑆)⟶𝒫 (𝑅 Cn 𝑆) → ran (𝑘 ∈ {𝑧 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑧) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) ⊆ 𝒫 (𝑅 Cn 𝑆))
8078, 79ax-mp 5 . . . . 5 ran (𝑘 ∈ {𝑧 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑧) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) ⊆ 𝒫 (𝑅 Cn 𝑆)
8177, 80ssexi 5279 . . . 4 ran (𝑘 ∈ {𝑧 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑧) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) ∈ V
8281a1i 11 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ran (𝑘 ∈ {𝑧 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑧) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) ∈ V)
83 topontop 22262 . . . 4 (𝑅 ∈ (TopOn‘𝑋) → 𝑅 ∈ Top)
84 topontop 22262 . . . 4 (𝑆 ∈ (TopOn‘𝑌) → 𝑆 ∈ Top)
854, 5, 6xkoval 22938 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑧 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑧) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))))
8683, 84, 85syl2an 596 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑆ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑧 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑧) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))))
87 eqid 2736 . . . . 5 (𝑆ko 𝑅) = (𝑆ko 𝑅)
8887xkotopon 22951 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
8983, 84, 88syl2an 596 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑆ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
9075, 82, 86, 89subbascn 22605 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ((𝑥𝑌 ↦ (𝑋 × {𝑥})) ∈ (𝑆 Cn (𝑆ko 𝑅)) ↔ ((𝑥𝑌 ↦ (𝑋 × {𝑥})):𝑌⟶(𝑅 Cn 𝑆) ∧ ∀𝑦 ∈ ran (𝑘 ∈ {𝑧 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑧) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})((𝑥𝑌 ↦ (𝑋 × {𝑥})) “ 𝑦) ∈ 𝑆)))
913, 74, 90mpbir2and 711 1 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑥𝑌 ↦ (𝑋 × {𝑥})) ∈ (𝑆 Cn (𝑆ko 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  {crab 3407  Vcvv 3445  cin 3909  wss 3910  c0 4282  ifcif 4486  𝒫 cpw 4560  {csn 4586   cuni 4865  cmpt 5188   × cxp 5631  ccnv 5632  ran crn 5634  cres 5635  cima 5636  wf 6492  cfv 6496  (class class class)co 7357  cmpo 7359  ficfi 9346  t crest 17302  topGenctg 17319  Topctop 22242  TopOnctopon 22259   Cn ccn 22575  Compccmp 22737  ko cxko 22912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-fin 8887  df-fi 9347  df-rest 17304  df-topgen 17325  df-top 22243  df-topon 22260  df-bases 22296  df-cn 22578  df-cnp 22579  df-cmp 22738  df-xko 22914
This theorem is referenced by:  cnmptkc  23030  xkofvcn  23035
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