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Theorem cnpfval 22720
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 22714 . . 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 770 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑗 = 𝐽)
43unieqd 4921 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑗 = 𝐽)
5 toponuni 22398 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
65ad2antrr 725 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑋 = 𝐽)
74, 6eqtr4d 2776 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑗 = 𝑋)
8 simprr 772 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑘 = 𝐾)
98unieqd 4921 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑘 = 𝐾)
10 toponuni 22398 . . . . . . 7 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
1110ad2antlr 726 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑌 = 𝐾)
129, 11eqtr4d 2776 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑘 = 𝑌)
1312, 7oveq12d 7422 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → ( 𝑘m 𝑗) = (𝑌m 𝑋))
143rexeqdv 3327 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → (∃𝑣𝑗 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤) ↔ ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤)))
1514imbi2d 341 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → (((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝑗 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤)) ↔ ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))))
168, 15raleqbidv 3343 . . . 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 3450 . . 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 5238 . 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 22397 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2019adantr 482 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → 𝐽 ∈ Top)
21 topontop 22397 . . 3 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
2221adantl 483 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → 𝐾 ∈ Top)
23 ovex 7437 . . . . . 6 (𝑌m 𝑋) ∈ V
24 ssrab2 4076 . . . . . 6 {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))} ⊆ (𝑌m 𝑋)
2523, 24elpwi2 5345 . . . . 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 7110 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥𝑋 ↦ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))}):𝑋⟶𝒫 (𝑌m 𝑋))
28 toponmax 22410 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
2928adantr 482 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → 𝑋𝐽)
3023pwex 5377 . . . 4 𝒫 (𝑌m 𝑋) ∈ V
3130a1i 11 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → 𝒫 (𝑌m 𝑋) ∈ V)
32 fex2 7919 . . 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 1372 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥𝑋 ↦ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))}) ∈ V)
342, 18, 20, 22, 33ovmpod 7555 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 397   = wceq 1542  wcel 2107  wral 3062  wrex 3071  {crab 3433  Vcvv 3475  wss 3947  𝒫 cpw 4601   cuni 4907  cmpt 5230  cima 5678  wf 6536  cfv 6540  (class class class)co 7404  cmpo 7406  m cmap 8816  Topctop 22377  TopOnctopon 22394   CnP ccnp 22711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7407  df-oprab 7408  df-mpo 7409  df-top 22378  df-topon 22395  df-cnp 22714
This theorem is referenced by:  cnpval  22722  iscnp2  22725  cnambfre  36474
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