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Theorem xkoccn 21470
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 (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.
StepHypRef Expression
1 cnconst2 21135 . . . 4 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌) ∧ 𝑥𝑌) → (𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆))
213expa 1284 . . 3 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑥𝑌) → (𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆))
3 eqid 2651 . . 3 (𝑥𝑌 ↦ (𝑋 × {𝑥})) = (𝑥𝑌 ↦ (𝑋 × {𝑥}))
42, 3fmptd 6425 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑥𝑌 ↦ (𝑋 × {𝑥})):𝑌⟶(𝑅 Cn 𝑆))
5 eqid 2651 . . . . . 6 𝑅 = 𝑅
6 eqid 2651 . . . . . 6 {𝑧 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑧) ∈ Comp} = {𝑧 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑧) ∈ Comp}
7 eqid 2651 . . . . . 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})
85, 6, 7xkobval 21437 . . . . 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})}
98abeq2i 2764 . . . 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}))
102adantlr 751 . . . . . . . . . . . . . 14 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ 𝑥𝑌) → (𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆))
1110adantlr 751 . . . . . . . . . . . . 13 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑥𝑌) → (𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆))
1211adantlr 751 . . . . . . . . . . . 12 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) ∧ 𝑥𝑌) → (𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆))
13 simplr 807 . . . . . . . . . . . . . 14 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) ∧ 𝑥𝑌) → 𝑘 = ∅)
1413imaeq2d 5501 . . . . . . . . . . . . 13 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) ∧ 𝑥𝑌) → ((𝑋 × {𝑥}) “ 𝑘) = ((𝑋 × {𝑥}) “ ∅))
15 ima0 5516 . . . . . . . . . . . . . 14 ((𝑋 × {𝑥}) “ ∅) = ∅
16 0ss 4005 . . . . . . . . . . . . . 14 ∅ ⊆ 𝑣
1715, 16eqsstri 3668 . . . . . . . . . . . . 13 ((𝑋 × {𝑥}) “ ∅) ⊆ 𝑣
1814, 17syl6eqss 3688 . . . . . . . . . . . 12 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) ∧ 𝑥𝑌) → ((𝑋 × {𝑥}) “ 𝑘) ⊆ 𝑣)
19 imaeq1 5496 . . . . . . . . . . . . . 14 (𝑓 = (𝑋 × {𝑥}) → (𝑓𝑘) = ((𝑋 × {𝑥}) “ 𝑘))
2019sseq1d 3665 . . . . . . . . . . . . 13 (𝑓 = (𝑋 × {𝑥}) → ((𝑓𝑘) ⊆ 𝑣 ↔ ((𝑋 × {𝑥}) “ 𝑘) ⊆ 𝑣))
2120elrab 3396 . . . . . . . . . . . 12 ((𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} ↔ ((𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆) ∧ ((𝑋 × {𝑥}) “ 𝑘) ⊆ 𝑣))
2212, 18, 21sylanbrc 699 . . . . . . . . . . 11 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) ∧ 𝑥𝑌) → (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})
2322ralrimiva 2995 . . . . . . . . . 10 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) → ∀𝑥𝑌 (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})
24 rabid2 3148 . . . . . . . . . 10 (𝑌 = {𝑥𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}} ↔ ∀𝑥𝑌 (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})
2523, 24sylibr 224 . . . . . . . . 9 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) → 𝑌 = {𝑥𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}})
26 simpllr 815 . . . . . . . . . . 11 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑆 ∈ (TopOn‘𝑌))
27 toponmax 20778 . . . . . . . . . . 11 (𝑆 ∈ (TopOn‘𝑌) → 𝑌𝑆)
2826, 27syl 17 . . . . . . . . . 10 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑌𝑆)
2928adantr 480 . . . . . . . . 9 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) → 𝑌𝑆)
3025, 29eqeltrrd 2731 . . . . . . . 8 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) → {𝑥𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}} ∈ 𝑆)
31 ifnefalse 4131 . . . . . . . . . . . . . . 15 (𝑘 ≠ ∅ → if(𝑘 = ∅, 𝑌, 𝑣) = 𝑣)
3231ad2antlr 763 . . . . . . . . . . . . . 14 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → if(𝑘 = ∅, 𝑌, 𝑣) = 𝑣)
3332eleq2d 2716 . . . . . . . . . . . . 13 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → (𝑥 ∈ if(𝑘 = ∅, 𝑌, 𝑣) ↔ 𝑥𝑣))
34 vex 3234 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
3534snss 4348 . . . . . . . . . . . . . . 15 (𝑥𝑣 ↔ {𝑥} ⊆ 𝑣)
3633, 35syl6bb 276 . . . . . . . . . . . . . 14 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → (𝑥 ∈ if(𝑘 = ∅, 𝑌, 𝑣) ↔ {𝑥} ⊆ 𝑣))
37 df-ima 5156 . . . . . . . . . . . . . . . . 17 ((𝑋 × {𝑥}) “ 𝑘) = ran ((𝑋 × {𝑥}) ↾ 𝑘)
38 simplrl 817 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑘 ∈ 𝒫 𝑅)
3938ad2antrr 762 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → 𝑘 ∈ 𝒫 𝑅)
4039elpwid 4203 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → 𝑘 𝑅)
41 toponuni 20767 . . . . . . . . . . . . . . . . . . . . 21 (𝑅 ∈ (TopOn‘𝑋) → 𝑋 = 𝑅)
4241ad5antr 773 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → 𝑋 = 𝑅)
4340, 42sseqtr4d 3675 . . . . . . . . . . . . . . . . . . 19 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → 𝑘𝑋)
44 xpssres 5469 . . . . . . . . . . . . . . . . . . 19 (𝑘𝑋 → ((𝑋 × {𝑥}) ↾ 𝑘) = (𝑘 × {𝑥}))
4543, 44syl 17 . . . . . . . . . . . . . . . . . 18 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → ((𝑋 × {𝑥}) ↾ 𝑘) = (𝑘 × {𝑥}))
4645rneqd 5385 . . . . . . . . . . . . . . . . 17 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → ran ((𝑋 × {𝑥}) ↾ 𝑘) = ran (𝑘 × {𝑥}))
4737, 46syl5eq 2697 . . . . . . . . . . . . . . . 16 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → ((𝑋 × {𝑥}) “ 𝑘) = ran (𝑘 × {𝑥}))
48 rnxp 5599 . . . . . . . . . . . . . . . . 17 (𝑘 ≠ ∅ → ran (𝑘 × {𝑥}) = {𝑥})
4948ad2antlr 763 . . . . . . . . . . . . . . . 16 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → ran (𝑘 × {𝑥}) = {𝑥})
5047, 49eqtrd 2685 . . . . . . . . . . . . . . 15 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → ((𝑋 × {𝑥}) “ 𝑘) = {𝑥})
5150sseq1d 3665 . . . . . . . . . . . . . 14 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → (((𝑋 × {𝑥}) “ 𝑘) ⊆ 𝑣 ↔ {𝑥} ⊆ 𝑣))
5211adantlr 751 . . . . . . . . . . . . . . 15 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → (𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆))
5352biantrurd 528 . . . . . . . . . . . . . 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)))
5436, 51, 533bitr2d 296 . . . . . . . . . . . . 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)))
5533, 54bitr3d 270 . . . . . . . . . . . 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)))
5655, 21syl6bbr 278 . . . . . . . . . . 11 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → (𝑥𝑣 ↔ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))
5756rabbi2dva 3854 . . . . . . . . . 10 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) → (𝑌𝑣) = {𝑥𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}})
58 simplrr 818 . . . . . . . . . . . . 13 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑣𝑆)
59 toponss 20779 . . . . . . . . . . . . 13 ((𝑆 ∈ (TopOn‘𝑌) ∧ 𝑣𝑆) → 𝑣𝑌)
6026, 58, 59syl2anc 694 . . . . . . . . . . . 12 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑣𝑌)
6160adantr 480 . . . . . . . . . . 11 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) → 𝑣𝑌)
62 sseqin2 3850 . . . . . . . . . . 11 (𝑣𝑌 ↔ (𝑌𝑣) = 𝑣)
6361, 62sylib 208 . . . . . . . . . 10 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) → (𝑌𝑣) = 𝑣)
6457, 63eqtr3d 2687 . . . . . . . . 9 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) → {𝑥𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}} = 𝑣)
6558adantr 480 . . . . . . . . 9 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) → 𝑣𝑆)
6664, 65eqeltrd 2730 . . . . . . . 8 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) → {𝑥𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}} ∈ 𝑆)
6730, 66pm2.61dane 2910 . . . . . . 7 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) → {𝑥𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}} ∈ 𝑆)
68 imaeq2 5497 . . . . . . . . 9 (𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} → ((𝑥𝑌 ↦ (𝑋 × {𝑥})) “ 𝑦) = ((𝑥𝑌 ↦ (𝑋 × {𝑥})) “ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))
693mptpreima 5666 . . . . . . . . 9 ((𝑥𝑌 ↦ (𝑋 × {𝑥})) “ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) = {𝑥𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}}
7068, 69syl6eq 2701 . . . . . . . 8 (𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} → ((𝑥𝑌 ↦ (𝑋 × {𝑥})) “ 𝑦) = {𝑥𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}})
7170eleq1d 2715 . . . . . . 7 (𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} → (((𝑥𝑌 ↦ (𝑋 × {𝑥})) “ 𝑦) ∈ 𝑆 ↔ {𝑥𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}} ∈ 𝑆))
7267, 71syl5ibrcom 237 . . . . . 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))
7372expimpd 628 . . . . 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))
7473rexlimdvva 3067 . . . 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))
759, 74syl5bi 232 . . 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))
7675ralrimiv 2994 . 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)
77 simpr 476 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑆 ∈ (TopOn‘𝑌))
78 ovex 6718 . . . . . 6 (𝑅 Cn 𝑆) ∈ V
7978pwex 4878 . . . . 5 𝒫 (𝑅 Cn 𝑆) ∈ V
805, 6, 7xkotf 21436 . . . . . 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)
81 frn 6091 . . . . . 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))
8280, 81ax-mp 5 . . . . 5 ran (𝑘 ∈ {𝑧 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑧) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) ⊆ 𝒫 (𝑅 Cn 𝑆)
8379, 82ssexi 4836 . . . 4 ran (𝑘 ∈ {𝑧 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑧) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) ∈ V
8483a1i 11 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ran (𝑘 ∈ {𝑧 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑧) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) ∈ V)
85 topontop 20766 . . . 4 (𝑅 ∈ (TopOn‘𝑋) → 𝑅 ∈ Top)
86 topontop 20766 . . . 4 (𝑆 ∈ (TopOn‘𝑌) → 𝑆 ∈ Top)
875, 6, 7xkoval 21438 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑧 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑧) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))))
8885, 86, 87syl2an 493 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑆 ^ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑧 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑧) ∈ Comp}, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))))
89 eqid 2651 . . . . 5 (𝑆 ^ko 𝑅) = (𝑆 ^ko 𝑅)
9089xkotopon 21451 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
9185, 86, 90syl2an 493 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑆 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
9277, 84, 88, 91subbascn 21106 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ((𝑥𝑌 ↦ (𝑋 × {𝑥})) ∈ (𝑆 Cn (𝑆 ^ko 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)))
934, 76, 92mpbir2and 977 1 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑥𝑌 ↦ (𝑋 × {𝑥})) ∈ (𝑆 Cn (𝑆 ^ko 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  cin 3606  wss 3607  c0 3948  ifcif 4119  𝒫 cpw 4191  {csn 4210   cuni 4468  cmpt 4762   × cxp 5141  ccnv 5142  ran crn 5144  cres 5145  cima 5146  wf 5922  cfv 5926  (class class class)co 6690  cmpt2 6692  ficfi 8357  t crest 16128  topGenctg 16145  Topctop 20746  TopOnctopon 20763   Cn ccn 21076  Compccmp 21237   ^ko cxko 21412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-fin 8001  df-fi 8358  df-rest 16130  df-topgen 16151  df-top 20747  df-topon 20764  df-bases 20798  df-cn 21079  df-cnp 21080  df-cmp 21238  df-xko 21414
This theorem is referenced by:  cnmptkc  21530  xkofvcn  21535
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