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Theorem cnpfval 13780
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 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐽 CnP 𝐾) = (π‘₯ ∈ 𝑋 ↦ {𝑓 ∈ (π‘Œ β†‘π‘š 𝑋) ∣ βˆ€π‘€ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝐽 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))}))
Distinct variable groups:   𝑀,𝑓,π‘₯,𝐾   𝑓,𝑋,𝑀,π‘₯   𝑓,π‘Œ,𝑀,π‘₯   𝑣,𝑓,𝐽,𝑀,π‘₯
Allowed substitution hints:   𝐾(𝑣)   𝑋(𝑣)   π‘Œ(𝑣)

Proof of Theorem cnpfval
Dummy variables 𝑗 π‘˜ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cnp 13774 . . 3 CnP = (𝑗 ∈ Top, π‘˜ ∈ Top ↦ (π‘₯ ∈ βˆͺ 𝑗 ↦ {𝑓 ∈ (βˆͺ π‘˜ β†‘π‘š βˆͺ 𝑗) ∣ βˆ€π‘€ ∈ π‘˜ ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝑗 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))}))
21a1i 9 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ CnP = (𝑗 ∈ Top, π‘˜ ∈ Top ↦ (π‘₯ ∈ βˆͺ 𝑗 ↦ {𝑓 ∈ (βˆͺ π‘˜ β†‘π‘š βˆͺ 𝑗) ∣ βˆ€π‘€ ∈ π‘˜ ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝑗 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))})))
3 simprl 529 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ 𝑗 = 𝐽)
43unieqd 3822 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ βˆͺ 𝑗 = βˆͺ 𝐽)
5 toponuni 13600 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
65ad2antrr 488 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ 𝑋 = βˆͺ 𝐽)
74, 6eqtr4d 2213 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ βˆͺ 𝑗 = 𝑋)
8 simprr 531 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ π‘˜ = 𝐾)
98unieqd 3822 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ βˆͺ π‘˜ = βˆͺ 𝐾)
10 toponuni 13600 . . . . . . 7 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ π‘Œ = βˆͺ 𝐾)
1110ad2antlr 489 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ π‘Œ = βˆͺ 𝐾)
129, 11eqtr4d 2213 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ βˆͺ π‘˜ = π‘Œ)
1312, 7oveq12d 5895 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ (βˆͺ π‘˜ β†‘π‘š βˆͺ 𝑗) = (π‘Œ β†‘π‘š 𝑋))
143rexeqdv 2680 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ (βˆƒπ‘£ ∈ 𝑗 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀) ↔ βˆƒπ‘£ ∈ 𝐽 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀)))
1514imbi2d 230 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ (((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝑗 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀)) ↔ ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝐽 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))))
168, 15raleqbidv 2685 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ (βˆ€π‘€ ∈ π‘˜ ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝑗 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀)) ↔ βˆ€π‘€ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝐽 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))))
1713, 16rabeqbidv 2734 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ {𝑓 ∈ (βˆͺ π‘˜ β†‘π‘š βˆͺ 𝑗) ∣ βˆ€π‘€ ∈ π‘˜ ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝑗 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))} = {𝑓 ∈ (π‘Œ β†‘π‘š 𝑋) ∣ βˆ€π‘€ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝐽 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))})
187, 17mpteq12dv 4087 . 2 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ π‘˜ = 𝐾)) β†’ (π‘₯ ∈ βˆͺ 𝑗 ↦ {𝑓 ∈ (βˆͺ π‘˜ β†‘π‘š βˆͺ 𝑗) ∣ βˆ€π‘€ ∈ π‘˜ ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝑗 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))}) = (π‘₯ ∈ 𝑋 ↦ {𝑓 ∈ (π‘Œ β†‘π‘š 𝑋) ∣ βˆ€π‘€ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝐽 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))}))
19 topontop 13599 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
2019adantr 276 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ 𝐽 ∈ Top)
21 topontop 13599 . . 3 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ 𝐾 ∈ Top)
2221adantl 277 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ 𝐾 ∈ Top)
23 fnmap 6657 . . . . . . . 8 β†‘π‘š Fn (V Γ— V)
2423a1i 9 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ β†‘π‘š Fn (V Γ— V))
25 toponmax 13610 . . . . . . . . 9 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ π‘Œ ∈ 𝐾)
2625elexd 2752 . . . . . . . 8 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ π‘Œ ∈ V)
2726adantl 277 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ π‘Œ ∈ V)
28 toponmax 13610 . . . . . . . . 9 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ 𝐽)
2928elexd 2752 . . . . . . . 8 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ V)
3029adantr 276 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ 𝑋 ∈ V)
31 fnovex 5910 . . . . . . 7 (( β†‘π‘š Fn (V Γ— V) ∧ π‘Œ ∈ V ∧ 𝑋 ∈ V) β†’ (π‘Œ β†‘π‘š 𝑋) ∈ V)
3224, 27, 30, 31syl3anc 1238 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (π‘Œ β†‘π‘š 𝑋) ∈ V)
3332adantr 276 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ π‘₯ ∈ 𝑋) β†’ (π‘Œ β†‘π‘š 𝑋) ∈ V)
34 ssrab2 3242 . . . . . 6 {𝑓 ∈ (π‘Œ β†‘π‘š 𝑋) ∣ βˆ€π‘€ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝐽 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))} βŠ† (π‘Œ β†‘π‘š 𝑋)
35 elpw2g 4158 . . . . . 6 ((π‘Œ β†‘π‘š 𝑋) ∈ V β†’ ({𝑓 ∈ (π‘Œ β†‘π‘š 𝑋) ∣ βˆ€π‘€ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝐽 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))} ∈ 𝒫 (π‘Œ β†‘π‘š 𝑋) ↔ {𝑓 ∈ (π‘Œ β†‘π‘š 𝑋) ∣ βˆ€π‘€ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝐽 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))} βŠ† (π‘Œ β†‘π‘š 𝑋)))
3634, 35mpbiri 168 . . . . 5 ((π‘Œ β†‘π‘š 𝑋) ∈ V β†’ {𝑓 ∈ (π‘Œ β†‘π‘š 𝑋) ∣ βˆ€π‘€ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝐽 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))} ∈ 𝒫 (π‘Œ β†‘π‘š 𝑋))
3733, 36syl 14 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ π‘₯ ∈ 𝑋) β†’ {𝑓 ∈ (π‘Œ β†‘π‘š 𝑋) ∣ βˆ€π‘€ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝐽 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))} ∈ 𝒫 (π‘Œ β†‘π‘š 𝑋))
3837fmpttd 5673 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (π‘₯ ∈ 𝑋 ↦ {𝑓 ∈ (π‘Œ β†‘π‘š 𝑋) ∣ βˆ€π‘€ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝐽 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))}):π‘‹βŸΆπ’« (π‘Œ β†‘π‘š 𝑋))
3928adantr 276 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ 𝑋 ∈ 𝐽)
4032pwexd 4183 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ 𝒫 (π‘Œ β†‘π‘š 𝑋) ∈ V)
41 fex2 5386 . . 3 (((π‘₯ ∈ 𝑋 ↦ {𝑓 ∈ (π‘Œ β†‘π‘š 𝑋) ∣ βˆ€π‘€ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝐽 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))}):π‘‹βŸΆπ’« (π‘Œ β†‘π‘š 𝑋) ∧ 𝑋 ∈ 𝐽 ∧ 𝒫 (π‘Œ β†‘π‘š 𝑋) ∈ V) β†’ (π‘₯ ∈ 𝑋 ↦ {𝑓 ∈ (π‘Œ β†‘π‘š 𝑋) ∣ βˆ€π‘€ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝐽 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))}) ∈ V)
4238, 39, 40, 41syl3anc 1238 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (π‘₯ ∈ 𝑋 ↦ {𝑓 ∈ (π‘Œ β†‘π‘š 𝑋) ∣ βˆ€π‘€ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝐽 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))}) ∈ V)
432, 18, 20, 22, 42ovmpod 6004 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐽 CnP 𝐾) = (π‘₯ ∈ 𝑋 ↦ {𝑓 ∈ (π‘Œ β†‘π‘š 𝑋) ∣ βˆ€π‘€ ∈ 𝐾 ((π‘“β€˜π‘₯) ∈ 𝑀 β†’ βˆƒπ‘£ ∈ 𝐽 (π‘₯ ∈ 𝑣 ∧ (𝑓 β€œ 𝑣) βŠ† 𝑀))}))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148  βˆ€wral 2455  βˆƒwrex 2456  {crab 2459  Vcvv 2739   βŠ† wss 3131  π’« cpw 3577  βˆͺ cuni 3811   ↦ cmpt 4066   Γ— cxp 4626   β€œ cima 4631   Fn wfn 5213  βŸΆwf 5214  β€˜cfv 5218  (class class class)co 5877   ∈ cmpo 5879   β†‘π‘š cmap 6650  Topctop 13582  TopOnctopon 13595   CnP ccnp 13771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-map 6652  df-top 13583  df-topon 13596  df-cnp 13774
This theorem is referenced by:  cnpval  13783
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