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Theorem xkoccn 22162
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 21826 . . . 4 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌) ∧ 𝑥𝑌) → (𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆))
213expa 1112 . . 3 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑥𝑌) → (𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆))
32fmpttd 6877 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑥𝑌 ↦ (𝑋 × {𝑥})):𝑌⟶(𝑅 Cn 𝑆))
4 eqid 2826 . . . . . 6 𝑅 = 𝑅
5 eqid 2826 . . . . . 6 {𝑧 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑧) ∈ Comp} = {𝑧 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑧) ∈ Comp}
6 eqid 2826 . . . . . 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 22129 . . . . 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})}
87abeq2i 2953 . . . 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 755 . . . . . . . . . . . 12 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) ∧ 𝑥𝑌) → (𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆))
10 simplr 765 . . . . . . . . . . . . . 14 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) ∧ 𝑥𝑌) → 𝑘 = ∅)
1110imaeq2d 5928 . . . . . . . . . . . . 13 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) ∧ 𝑥𝑌) → ((𝑋 × {𝑥}) “ 𝑘) = ((𝑋 × {𝑥}) “ ∅))
12 ima0 5944 . . . . . . . . . . . . . 14 ((𝑋 × {𝑥}) “ ∅) = ∅
13 0ss 4354 . . . . . . . . . . . . . 14 ∅ ⊆ 𝑣
1412, 13eqsstri 4005 . . . . . . . . . . . . 13 ((𝑋 × {𝑥}) “ ∅) ⊆ 𝑣
1511, 14eqsstrdi 4025 . . . . . . . . . . . 12 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) ∧ 𝑥𝑌) → ((𝑋 × {𝑥}) “ 𝑘) ⊆ 𝑣)
16 imaeq1 5923 . . . . . . . . . . . . . 14 (𝑓 = (𝑋 × {𝑥}) → (𝑓𝑘) = ((𝑋 × {𝑥}) “ 𝑘))
1716sseq1d 4002 . . . . . . . . . . . . 13 (𝑓 = (𝑋 × {𝑥}) → ((𝑓𝑘) ⊆ 𝑣 ↔ ((𝑋 × {𝑥}) “ 𝑘) ⊆ 𝑣))
1817elrab 3684 . . . . . . . . . . . 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 3187 . . . . . . . . . 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 3387 . . . . . . . . . 10 (𝑌 = {𝑥𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}} ↔ ∀𝑥𝑌 (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})
2220, 21sylibr 235 . . . . . . . . 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 772 . . . . . . . . . . 11 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑆 ∈ (TopOn‘𝑌))
24 toponmax 21469 . . . . . . . . . . 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 2919 . . . . . . . 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 4482 . . . . . . . . . . . . . . 15 (𝑘 ≠ ∅ → if(𝑘 = ∅, 𝑌, 𝑣) = 𝑣)
2928ad2antlr 723 . . . . . . . . . . . . . 14 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → if(𝑘 = ∅, 𝑌, 𝑣) = 𝑣)
3029eleq2d 2903 . . . . . . . . . . . . 13 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → (𝑥 ∈ if(𝑘 = ∅, 𝑌, 𝑣) ↔ 𝑥𝑣))
31 vex 3503 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
3231snss 4717 . . . . . . . . . . . . . . 15 (𝑥𝑣 ↔ {𝑥} ⊆ 𝑣)
3330, 32syl6bb 288 . . . . . . . . . . . . . 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 5567 . . . . . . . . . . . . . . . . 17 ((𝑋 × {𝑥}) “ 𝑘) = ran ((𝑋 × {𝑥}) ↾ 𝑘)
35 simplrl 773 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑘 ∈ 𝒫 𝑅)
3635ad2antrr 722 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → 𝑘 ∈ 𝒫 𝑅)
3736elpwid 4556 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → 𝑘 𝑅)
38 toponuni 21457 . . . . . . . . . . . . . . . . . . . . 21 (𝑅 ∈ (TopOn‘𝑋) → 𝑋 = 𝑅)
3938ad5antr 730 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → 𝑋 = 𝑅)
4037, 39sseqtrrd 4012 . . . . . . . . . . . . . . . . . . 19 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → 𝑘𝑋)
41 xpssres 5888 . . . . . . . . . . . . . . . . . . 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 5807 . . . . . . . . . . . . . . . . 17 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → ran ((𝑋 × {𝑥}) ↾ 𝑘) = ran (𝑘 × {𝑥}))
4434, 43syl5eq 2873 . . . . . . . . . . . . . . . 16 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → ((𝑋 × {𝑥}) “ 𝑘) = ran (𝑘 × {𝑥}))
45 rnxp 6026 . . . . . . . . . . . . . . . . 17 (𝑘 ≠ ∅ → ran (𝑘 × {𝑥}) = {𝑥})
4645ad2antlr 723 . . . . . . . . . . . . . . . 16 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → ran (𝑘 × {𝑥}) = {𝑥})
4744, 46eqtrd 2861 . . . . . . . . . . . . . . 15 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → ((𝑋 × {𝑥}) “ 𝑘) = {𝑥})
4847sseq1d 4002 . . . . . . . . . . . . . 14 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → (((𝑋 × {𝑥}) “ 𝑘) ⊆ 𝑣 ↔ {𝑥} ⊆ 𝑣))
492ad5ant15 755 . . . . . . . . . . . . . . 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 308 . . . . . . . . . . . . 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 282 . . . . . . . . . . . 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, 18syl6bbr 290 . . . . . . . . . . 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 4198 . . . . . . . . . 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 774 . . . . . . . . . . . . 13 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑣𝑆)
56 toponss 21470 . . . . . . . . . . . . 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 4196 . . . . . . . . . . 11 (𝑣𝑌 ↔ (𝑌𝑣) = 𝑣)
6058, 59sylib 219 . . . . . . . . . 10 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) → (𝑌𝑣) = 𝑣)
6154, 60eqtr3d 2863 . . . . . . . . 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 2918 . . . . . . . 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 3109 . . . . . . 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 5924 . . . . . . . . 9 (𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} → ((𝑥𝑌 ↦ (𝑋 × {𝑥})) “ 𝑦) = ((𝑥𝑌 ↦ (𝑋 × {𝑥})) “ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))
66 eqid 2826 . . . . . . . . . 10 (𝑥𝑌 ↦ (𝑋 × {𝑥})) = (𝑥𝑌 ↦ (𝑋 × {𝑥}))
6766mptpreima 6091 . . . . . . . . 9 ((𝑥𝑌 ↦ (𝑋 × {𝑥})) “ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) = {𝑥𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}}
6865, 67syl6eq 2877 . . . . . . . 8 (𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} → ((𝑥𝑌 ↦ (𝑋 × {𝑥})) “ 𝑦) = {𝑥𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}})
6968eleq1d 2902 . . . . . . 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 248 . . . . . 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 3299 . . . 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, 72syl5bi 243 . . 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 3186 . 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 7183 . . . . . 6 (𝑅 Cn 𝑆) ∈ V
7776pwex 5278 . . . . 5 𝒫 (𝑅 Cn 𝑆) ∈ V
784, 5, 6xkotf 22128 . . . . . 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 6519 . . . . . 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 5223 . . . 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 21456 . . . 4 (𝑅 ∈ (TopOn‘𝑋) → 𝑅 ∈ Top)
84 topontop 21456 . . . 4 (𝑆 ∈ (TopOn‘𝑌) → 𝑆 ∈ Top)
854, 5, 6xkoval 22130 . . . 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 595 . . 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 2826 . . . . 5 (𝑆ko 𝑅) = (𝑆ko 𝑅)
8887xkotopon 22143 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
8983, 84, 88syl2an 595 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑆ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
9075, 82, 86, 89subbascn 21797 . 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 709 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 1530  wcel 2107  wne 3021  wral 3143  wrex 3144  {crab 3147  Vcvv 3500  cin 3939  wss 3940  c0 4295  ifcif 4470  𝒫 cpw 4542  {csn 4564   cuni 4837  cmpt 5143   × cxp 5552  ccnv 5553  ran crn 5555  cres 5556  cima 5557  wf 6350  cfv 6354  (class class class)co 7150  cmpo 7152  ficfi 8868  t crest 16689  topGenctg 16706  Topctop 21436  TopOnctopon 21453   Cn ccn 21767  Compccmp 21929  ko cxko 22104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-iin 4920  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-1st 7685  df-2nd 7686  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8284  df-map 8403  df-en 8504  df-dom 8505  df-fin 8507  df-fi 8869  df-rest 16691  df-topgen 16712  df-top 21437  df-topon 21454  df-bases 21489  df-cn 21770  df-cnp 21771  df-cmp 21930  df-xko 22106
This theorem is referenced by:  cnmptkc  22222  xkofvcn  22227
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