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Theorem cnpfval 12401
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 12395 . . 3 CnP = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ (𝑥 𝑗 ↦ {𝑓 ∈ ( 𝑘𝑚 𝑗) ∣ ∀𝑤𝑘 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝑗 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))}))
21a1i 9 . 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 521 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑗 = 𝐽)
43unieqd 3754 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑗 = 𝐽)
5 toponuni 12219 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
65ad2antrr 480 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑋 = 𝐽)
74, 6eqtr4d 2176 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑗 = 𝑋)
8 simprr 522 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑘 = 𝐾)
98unieqd 3754 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑘 = 𝐾)
10 toponuni 12219 . . . . . . 7 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
1110ad2antlr 481 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑌 = 𝐾)
129, 11eqtr4d 2176 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → 𝑘 = 𝑌)
1312, 7oveq12d 5799 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → ( 𝑘𝑚 𝑗) = (𝑌𝑚 𝑋))
143rexeqdv 2636 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → (∃𝑣𝑗 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤) ↔ ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤)))
1514imbi2d 229 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑗 = 𝐽𝑘 = 𝐾)) → (((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝑗 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤)) ↔ ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))))
168, 15raleqbidv 2641 . . . 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 2684 . . 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 4017 . 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 12218 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2019adantr 274 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → 𝐽 ∈ Top)
21 topontop 12218 . . 3 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
2221adantl 275 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → 𝐾 ∈ Top)
23 fnmap 6556 . . . . . . . 8 𝑚 Fn (V × V)
2423a1i 9 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ↑𝑚 Fn (V × V))
25 toponmax 12229 . . . . . . . . 9 (𝐾 ∈ (TopOn‘𝑌) → 𝑌𝐾)
2625elexd 2702 . . . . . . . 8 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 ∈ V)
2726adantl 275 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → 𝑌 ∈ V)
28 toponmax 12229 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
2928elexd 2702 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ V)
3029adantr 274 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → 𝑋 ∈ V)
31 fnovex 5811 . . . . . . 7 (( ↑𝑚 Fn (V × V) ∧ 𝑌 ∈ V ∧ 𝑋 ∈ V) → (𝑌𝑚 𝑋) ∈ V)
3224, 27, 30, 31syl3anc 1217 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑌𝑚 𝑋) ∈ V)
3332adantr 274 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥𝑋) → (𝑌𝑚 𝑋) ∈ V)
34 ssrab2 3186 . . . . . 6 {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))} ⊆ (𝑌𝑚 𝑋)
35 elpw2g 4088 . . . . . 6 ((𝑌𝑚 𝑋) ∈ V → ({𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))} ∈ 𝒫 (𝑌𝑚 𝑋) ↔ {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))} ⊆ (𝑌𝑚 𝑋)))
3634, 35mpbiri 167 . . . . 5 ((𝑌𝑚 𝑋) ∈ V → {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))} ∈ 𝒫 (𝑌𝑚 𝑋))
3733, 36syl 14 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑥𝑋) → {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))} ∈ 𝒫 (𝑌𝑚 𝑋))
3837fmpttd 5582 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥𝑋 ↦ {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))}):𝑋⟶𝒫 (𝑌𝑚 𝑋))
3928adantr 274 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → 𝑋𝐽)
4032pwexd 4112 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → 𝒫 (𝑌𝑚 𝑋) ∈ V)
41 fex2 5298 . . 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)
4238, 39, 40, 41syl3anc 1217 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥𝑋 ↦ {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))}) ∈ V)
432, 18, 20, 22, 42ovmpod 5905 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 set class
Syntax hints:  wi 4  wa 103   = wceq 1332  wcel 1481  wral 2417  wrex 2418  {crab 2421  Vcvv 2689  wss 3075  𝒫 cpw 3514   cuni 3743  cmpt 3996   × cxp 4544  cima 4549   Fn wfn 5125  wf 5126  cfv 5130  (class class class)co 5781  cmpo 5783  𝑚 cmap 6549  Topctop 12201  TopOnctopon 12214   CnP ccnp 12392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-map 6551  df-top 12202  df-topon 12215  df-cnp 12395
This theorem is referenced by:  cnpval  12404
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