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Theorem cnpfval 21770
Description: The function mapping the points in a topology 𝐽 to the set of all functions from 𝐽 to topology 𝐾 continuous at that point. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
cnpfval ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 CnP 𝐾) = (𝑥𝑋 ↦ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))}))
Distinct variable groups:   𝑤,𝑓,𝑥,𝐾   𝑓,𝑋,𝑤,𝑥   𝑓,𝑌,𝑤,𝑥   𝑣,𝑓,𝐽,𝑤,𝑥
Allowed substitution hints:   𝐾(𝑣)   𝑋(𝑣)   𝑌(𝑣)

Proof of Theorem cnpfval
Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cnp 21764 . . 3 CnP = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ (𝑥 𝑗 ↦ {𝑓 ∈ ( 𝑘m 𝑗) ∣ ∀𝑤𝑘 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝑗 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))}))
21a1i 11 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → CnP = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ (𝑥 𝑗 ↦ {𝑓 ∈ ( 𝑘m 𝑗) ∣ ∀𝑤𝑘 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝑗 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))})))
3 simprl 767 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑗 = 𝐽)
43unieqd 4840 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑗 = 𝐽)
5 toponuni 21450 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
65ad2antrr 722 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑋 = 𝐽)
74, 6eqtr4d 2856 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑗 = 𝑋)
8 simprr 769 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑘 = 𝐾)
98unieqd 4840 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑘 = 𝐾)
10 toponuni 21450 . . . . . . 7 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
1110ad2antlr 723 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑌 = 𝐾)
129, 11eqtr4d 2856 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑘 = 𝑌)
1312, 7oveq12d 7163 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → ( 𝑘m 𝑗) = (𝑌m 𝑋))
143rexeqdv 3414 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → (∃𝑣𝑗 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤) ↔ ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤)))
1514imbi2d 342 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → (((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝑗 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤)) ↔ ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))))
168, 15raleqbidv 3399 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → (∀𝑤𝑘 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝑗 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤)) ↔ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))))
1713, 16rabeqbidv 3483 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → {𝑓 ∈ ( 𝑘m 𝑗) ∣ ∀𝑤𝑘 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝑗 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))} = {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))})
187, 17mpteq12dv 5142 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → (𝑥 𝑗 ↦ {𝑓 ∈ ( 𝑘m 𝑗) ∣ ∀𝑤𝑘 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝑗 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))}) = (𝑥𝑋 ↦ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))}))
19 topontop 21449 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2019adantr 481 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → 𝐽 ∈ Top)
21 topontop 21449 . . 3 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
2221adantl 482 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → 𝐾 ∈ Top)
23 ovex 7178 . . . . . 6 (𝑌m 𝑋) ∈ V
24 ssrab2 4053 . . . . . 6 {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))} ⊆ (𝑌m 𝑋)
2523, 24elpwi2 5240 . . . . 5 {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))} ∈ 𝒫 (𝑌m 𝑋)
2625a1i 11 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥𝑋) → {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))} ∈ 𝒫 (𝑌m 𝑋))
2726fmpttd 6871 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥𝑋 ↦ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))}):𝑋⟶𝒫 (𝑌m 𝑋))
28 toponmax 21462 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
2928adantr 481 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → 𝑋𝐽)
3023pwex 5272 . . . 4 𝒫 (𝑌m 𝑋) ∈ V
3130a1i 11 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → 𝒫 (𝑌m 𝑋) ∈ V)
32 fex2 7627 . . 3 (((𝑥𝑋 ↦ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))}):𝑋⟶𝒫 (𝑌m 𝑋) ∧ 𝑋𝐽 ∧ 𝒫 (𝑌m 𝑋) ∈ V) → (𝑥𝑋 ↦ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))}) ∈ V)
3327, 29, 31, 32syl3anc 1363 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥𝑋 ↦ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))}) ∈ V)
342, 18, 20, 22, 33ovmpod 7291 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 CnP 𝐾) = (𝑥𝑋 ↦ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  wrex 3136  {crab 3139  Vcvv 3492  wss 3933  𝒫 cpw 4535   cuni 4830  cmpt 5137  cima 5551  wf 6344  cfv 6348  (class class class)co 7145  cmpo 7147  m cmap 8395  Topctop 21429  TopOnctopon 21446   CnP ccnp 21761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-top 21430  df-topon 21447  df-cnp 21764
This theorem is referenced by:  cnpval  21772  iscnp2  21775  cnambfre  34821
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