MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xkoccn Structured version   Visualization version   GIF version

Theorem xkoccn 21702
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 21367 . . . 4 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌) ∧ 𝑥𝑌) → (𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆))
213expa 1147 . . 3 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑥𝑌) → (𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆))
32fmpttd 6575 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑥𝑌 ↦ (𝑋 × {𝑥})):𝑌⟶(𝑅 Cn 𝑆))
4 eqid 2765 . . . . . 6 𝑅 = 𝑅
5 eqid 2765 . . . . . 6 {𝑧 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑧) ∈ Comp} = {𝑧 ∈ 𝒫 𝑅 ∣ (𝑅t 𝑧) ∈ Comp}
6 eqid 2765 . . . . . 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 21669 . . . . 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 2878 . . . 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}))
92adantlr 706 . . . . . . . . . . . . . 14 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ 𝑥𝑌) → (𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆))
109adantlr 706 . . . . . . . . . . . . 13 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑥𝑌) → (𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆))
1110adantlr 706 . . . . . . . . . . . 12 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) ∧ 𝑥𝑌) → (𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆))
12 simplr 785 . . . . . . . . . . . . . 14 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) ∧ 𝑥𝑌) → 𝑘 = ∅)
1312imaeq2d 5648 . . . . . . . . . . . . 13 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) ∧ 𝑥𝑌) → ((𝑋 × {𝑥}) “ 𝑘) = ((𝑋 × {𝑥}) “ ∅))
14 ima0 5663 . . . . . . . . . . . . . 14 ((𝑋 × {𝑥}) “ ∅) = ∅
15 0ss 4134 . . . . . . . . . . . . . 14 ∅ ⊆ 𝑣
1614, 15eqsstri 3795 . . . . . . . . . . . . 13 ((𝑋 × {𝑥}) “ ∅) ⊆ 𝑣
1713, 16syl6eqss 3815 . . . . . . . . . . . 12 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) ∧ 𝑥𝑌) → ((𝑋 × {𝑥}) “ 𝑘) ⊆ 𝑣)
18 imaeq1 5643 . . . . . . . . . . . . . 14 (𝑓 = (𝑋 × {𝑥}) → (𝑓𝑘) = ((𝑋 × {𝑥}) “ 𝑘))
1918sseq1d 3792 . . . . . . . . . . . . 13 (𝑓 = (𝑋 × {𝑥}) → ((𝑓𝑘) ⊆ 𝑣 ↔ ((𝑋 × {𝑥}) “ 𝑘) ⊆ 𝑣))
2019elrab 3519 . . . . . . . . . . . 12 ((𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} ↔ ((𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆) ∧ ((𝑋 × {𝑥}) “ 𝑘) ⊆ 𝑣))
2111, 17, 20sylanbrc 578 . . . . . . . . . . 11 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) ∧ 𝑥𝑌) → (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})
2221ralrimiva 3113 . . . . . . . . . 10 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) → ∀𝑥𝑌 (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})
23 rabid2 3266 . . . . . . . . . 10 (𝑌 = {𝑥𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}} ↔ ∀𝑥𝑌 (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})
2422, 23sylibr 225 . . . . . . . . 9 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) → 𝑌 = {𝑥𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}})
25 simpllr 793 . . . . . . . . . . 11 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑆 ∈ (TopOn‘𝑌))
26 toponmax 21010 . . . . . . . . . . 11 (𝑆 ∈ (TopOn‘𝑌) → 𝑌𝑆)
2725, 26syl 17 . . . . . . . . . 10 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑌𝑆)
2827adantr 472 . . . . . . . . 9 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) → 𝑌𝑆)
2924, 28eqeltrrd 2845 . . . . . . . 8 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 = ∅) → {𝑥𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}} ∈ 𝑆)
30 ifnefalse 4255 . . . . . . . . . . . . . . 15 (𝑘 ≠ ∅ → if(𝑘 = ∅, 𝑌, 𝑣) = 𝑣)
3130ad2antlr 718 . . . . . . . . . . . . . 14 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → if(𝑘 = ∅, 𝑌, 𝑣) = 𝑣)
3231eleq2d 2830 . . . . . . . . . . . . 13 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → (𝑥 ∈ if(𝑘 = ∅, 𝑌, 𝑣) ↔ 𝑥𝑣))
33 vex 3353 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
3433snss 4470 . . . . . . . . . . . . . . 15 (𝑥𝑣 ↔ {𝑥} ⊆ 𝑣)
3532, 34syl6bb 278 . . . . . . . . . . . . . 14 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → (𝑥 ∈ if(𝑘 = ∅, 𝑌, 𝑣) ↔ {𝑥} ⊆ 𝑣))
36 df-ima 5290 . . . . . . . . . . . . . . . . 17 ((𝑋 × {𝑥}) “ 𝑘) = ran ((𝑋 × {𝑥}) ↾ 𝑘)
37 simplrl 795 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑘 ∈ 𝒫 𝑅)
3837ad2antrr 717 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → 𝑘 ∈ 𝒫 𝑅)
3938elpwid 4327 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → 𝑘 𝑅)
40 toponuni 20998 . . . . . . . . . . . . . . . . . . . . 21 (𝑅 ∈ (TopOn‘𝑋) → 𝑋 = 𝑅)
4140ad5antr 728 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → 𝑋 = 𝑅)
4239, 41sseqtr4d 3802 . . . . . . . . . . . . . . . . . . 19 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → 𝑘𝑋)
43 xpssres 5608 . . . . . . . . . . . . . . . . . . 19 (𝑘𝑋 → ((𝑋 × {𝑥}) ↾ 𝑘) = (𝑘 × {𝑥}))
4442, 43syl 17 . . . . . . . . . . . . . . . . . 18 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → ((𝑋 × {𝑥}) ↾ 𝑘) = (𝑘 × {𝑥}))
4544rneqd 5521 . . . . . . . . . . . . . . . . 17 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → ran ((𝑋 × {𝑥}) ↾ 𝑘) = ran (𝑘 × {𝑥}))
4636, 45syl5eq 2811 . . . . . . . . . . . . . . . 16 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → ((𝑋 × {𝑥}) “ 𝑘) = ran (𝑘 × {𝑥}))
47 rnxp 5747 . . . . . . . . . . . . . . . . 17 (𝑘 ≠ ∅ → ran (𝑘 × {𝑥}) = {𝑥})
4847ad2antlr 718 . . . . . . . . . . . . . . . 16 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → ran (𝑘 × {𝑥}) = {𝑥})
4946, 48eqtrd 2799 . . . . . . . . . . . . . . 15 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → ((𝑋 × {𝑥}) “ 𝑘) = {𝑥})
5049sseq1d 3792 . . . . . . . . . . . . . 14 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → (((𝑋 × {𝑥}) “ 𝑘) ⊆ 𝑣 ↔ {𝑥} ⊆ 𝑣))
5110adantlr 706 . . . . . . . . . . . . . . 15 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → (𝑋 × {𝑥}) ∈ (𝑅 Cn 𝑆))
5251biantrurd 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)))
5335, 50, 523bitr2d 298 . . . . . . . . . . . . 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)))
5432, 53bitr3d 272 . . . . . . . . . . . 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)))
5554, 20syl6bbr 280 . . . . . . . . . . 11 ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) ∧ 𝑥𝑌) → (𝑥𝑣 ↔ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))
5655rabbi2dva 3981 . . . . . . . . . 10 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) → (𝑌𝑣) = {𝑥𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}})
57 simplrr 796 . . . . . . . . . . . . 13 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑣𝑆)
58 toponss 21011 . . . . . . . . . . . . 13 ((𝑆 ∈ (TopOn‘𝑌) ∧ 𝑣𝑆) → 𝑣𝑌)
5925, 57, 58syl2anc 579 . . . . . . . . . . . 12 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) → 𝑣𝑌)
6059adantr 472 . . . . . . . . . . 11 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) → 𝑣𝑌)
61 sseqin2 3979 . . . . . . . . . . 11 (𝑣𝑌 ↔ (𝑌𝑣) = 𝑣)
6260, 61sylib 209 . . . . . . . . . 10 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) → (𝑌𝑣) = 𝑣)
6356, 62eqtr3d 2801 . . . . . . . . 9 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) → {𝑥𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}} = 𝑣)
6457adantr 472 . . . . . . . . 9 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) → 𝑣𝑆)
6563, 64eqeltrd 2844 . . . . . . . 8 (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) ∧ 𝑘 ≠ ∅) → {𝑥𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}} ∈ 𝑆)
6629, 65pm2.61dane 3024 . . . . . . 7 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 𝑅𝑣𝑆)) ∧ (𝑅t 𝑘) ∈ Comp) → {𝑥𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}} ∈ 𝑆)
67 imaeq2 5644 . . . . . . . . 9 (𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} → ((𝑥𝑌 ↦ (𝑋 × {𝑥})) “ 𝑦) = ((𝑥𝑌 ↦ (𝑋 × {𝑥})) “ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}))
68 eqid 2765 . . . . . . . . . 10 (𝑥𝑌 ↦ (𝑋 × {𝑥})) = (𝑥𝑌 ↦ (𝑋 × {𝑥}))
6968mptpreima 5814 . . . . . . . . 9 ((𝑥𝑌 ↦ (𝑋 × {𝑥})) “ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}) = {𝑥𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}}
7067, 69syl6eq 2815 . . . . . . . 8 (𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} → ((𝑥𝑌 ↦ (𝑋 × {𝑥})) “ 𝑦) = {𝑥𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}})
7170eleq1d 2829 . . . . . . 7 (𝑦 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣} → (((𝑥𝑌 ↦ (𝑋 × {𝑥})) “ 𝑦) ∈ 𝑆 ↔ {𝑥𝑌 ∣ (𝑋 × {𝑥}) ∈ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣}} ∈ 𝑆))
7266, 71syl5ibrcom 238 . . . . . 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 445 . . . . 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 3185 . . . 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))
758, 74syl5bi 233 . . 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 3112 . 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 477 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑆 ∈ (TopOn‘𝑌))
78 ovex 6874 . . . . . 6 (𝑅 Cn 𝑆) ∈ V
7978pwex 5016 . . . . 5 𝒫 (𝑅 Cn 𝑆) ∈ V
804, 5, 6xkotf 21668 . . . . . 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 6229 . . . . . 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 4964 . . . 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 20997 . . . 4 (𝑅 ∈ (TopOn‘𝑋) → 𝑅 ∈ Top)
86 topontop 20997 . . . 4 (𝑆 ∈ (TopOn‘𝑌) → 𝑆 ∈ Top)
874, 5, 6xkoval 21670 . . . 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 589 . . 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 2765 . . . . 5 (𝑆 ^ko 𝑅) = (𝑆 ^ko 𝑅)
9089xkotopon 21683 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
9185, 86, 90syl2an 589 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑆 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
9277, 84, 88, 91subbascn 21338 . 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)))
933, 76, 92mpbir2and 704 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 384   = wceq 1652  wcel 2155  wne 2937  wral 3055  wrex 3056  {crab 3059  Vcvv 3350  cin 3731  wss 3732  c0 4079  ifcif 4243  𝒫 cpw 4315  {csn 4334   cuni 4594  cmpt 4888   × cxp 5275  ccnv 5276  ran crn 5278  cres 5279  cima 5280  wf 6064  cfv 6068  (class class class)co 6842  cmpt2 6844  ficfi 8523  t crest 16349  topGenctg 16366  Topctop 20977  TopOnctopon 20994   Cn ccn 21308  Compccmp 21469   ^ko cxko 21644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-map 8062  df-en 8161  df-dom 8162  df-fin 8164  df-fi 8524  df-rest 16351  df-topgen 16372  df-top 20978  df-topon 20995  df-bases 21030  df-cn 21311  df-cnp 21312  df-cmp 21470  df-xko 21646
This theorem is referenced by:  cnmptkc  21762  xkofvcn  21767
  Copyright terms: Public domain W3C validator