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Theorem cnpval 22603
Description: The set of all functions from topology 𝐽 to topology 𝐾 that are continuous at a point 𝑃. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
Assertion
Ref Expression
cnpval ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) β†’ ((𝐽 CnP 𝐾)β€˜π‘ƒ) = {𝑓 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘¦ ∈ 𝐾 ((π‘“β€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝑓 β€œ π‘₯) βŠ† 𝑦))})
Distinct variable groups:   π‘₯,𝑓,𝑦,𝐽   𝑓,𝐾,π‘₯,𝑦   𝑓,𝑋,π‘₯,𝑦   𝑃,𝑓,π‘₯,𝑦   𝑓,π‘Œ,π‘₯,𝑦

Proof of Theorem cnpval
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 cnpfval 22601 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐽 CnP 𝐾) = (𝑣 ∈ 𝑋 ↦ {𝑓 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘¦ ∈ 𝐾 ((π‘“β€˜π‘£) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑣 ∈ π‘₯ ∧ (𝑓 β€œ π‘₯) βŠ† 𝑦))}))
21fveq1d 6849 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ ((𝐽 CnP 𝐾)β€˜π‘ƒ) = ((𝑣 ∈ 𝑋 ↦ {𝑓 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘¦ ∈ 𝐾 ((π‘“β€˜π‘£) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑣 ∈ π‘₯ ∧ (𝑓 β€œ π‘₯) βŠ† 𝑦))})β€˜π‘ƒ))
3 fveq2 6847 . . . . . . . 8 (𝑣 = 𝑃 β†’ (π‘“β€˜π‘£) = (π‘“β€˜π‘ƒ))
43eleq1d 2823 . . . . . . 7 (𝑣 = 𝑃 β†’ ((π‘“β€˜π‘£) ∈ 𝑦 ↔ (π‘“β€˜π‘ƒ) ∈ 𝑦))
5 eleq1 2826 . . . . . . . . 9 (𝑣 = 𝑃 β†’ (𝑣 ∈ π‘₯ ↔ 𝑃 ∈ π‘₯))
65anbi1d 631 . . . . . . . 8 (𝑣 = 𝑃 β†’ ((𝑣 ∈ π‘₯ ∧ (𝑓 β€œ π‘₯) βŠ† 𝑦) ↔ (𝑃 ∈ π‘₯ ∧ (𝑓 β€œ π‘₯) βŠ† 𝑦)))
76rexbidv 3176 . . . . . . 7 (𝑣 = 𝑃 β†’ (βˆƒπ‘₯ ∈ 𝐽 (𝑣 ∈ π‘₯ ∧ (𝑓 β€œ π‘₯) βŠ† 𝑦) ↔ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝑓 β€œ π‘₯) βŠ† 𝑦)))
84, 7imbi12d 345 . . . . . 6 (𝑣 = 𝑃 β†’ (((π‘“β€˜π‘£) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑣 ∈ π‘₯ ∧ (𝑓 β€œ π‘₯) βŠ† 𝑦)) ↔ ((π‘“β€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝑓 β€œ π‘₯) βŠ† 𝑦))))
98ralbidv 3175 . . . . 5 (𝑣 = 𝑃 β†’ (βˆ€π‘¦ ∈ 𝐾 ((π‘“β€˜π‘£) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑣 ∈ π‘₯ ∧ (𝑓 β€œ π‘₯) βŠ† 𝑦)) ↔ βˆ€π‘¦ ∈ 𝐾 ((π‘“β€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝑓 β€œ π‘₯) βŠ† 𝑦))))
109rabbidv 3418 . . . 4 (𝑣 = 𝑃 β†’ {𝑓 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘¦ ∈ 𝐾 ((π‘“β€˜π‘£) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑣 ∈ π‘₯ ∧ (𝑓 β€œ π‘₯) βŠ† 𝑦))} = {𝑓 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘¦ ∈ 𝐾 ((π‘“β€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝑓 β€œ π‘₯) βŠ† 𝑦))})
11 eqid 2737 . . . 4 (𝑣 ∈ 𝑋 ↦ {𝑓 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘¦ ∈ 𝐾 ((π‘“β€˜π‘£) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑣 ∈ π‘₯ ∧ (𝑓 β€œ π‘₯) βŠ† 𝑦))}) = (𝑣 ∈ 𝑋 ↦ {𝑓 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘¦ ∈ 𝐾 ((π‘“β€˜π‘£) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑣 ∈ π‘₯ ∧ (𝑓 β€œ π‘₯) βŠ† 𝑦))})
12 ovex 7395 . . . . 5 (π‘Œ ↑m 𝑋) ∈ V
1312rabex 5294 . . . 4 {𝑓 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘¦ ∈ 𝐾 ((π‘“β€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝑓 β€œ π‘₯) βŠ† 𝑦))} ∈ V
1410, 11, 13fvmpt 6953 . . 3 (𝑃 ∈ 𝑋 β†’ ((𝑣 ∈ 𝑋 ↦ {𝑓 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘¦ ∈ 𝐾 ((π‘“β€˜π‘£) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑣 ∈ π‘₯ ∧ (𝑓 β€œ π‘₯) βŠ† 𝑦))})β€˜π‘ƒ) = {𝑓 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘¦ ∈ 𝐾 ((π‘“β€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝑓 β€œ π‘₯) βŠ† 𝑦))})
152, 14sylan9eq 2797 . 2 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝑃 ∈ 𝑋) β†’ ((𝐽 CnP 𝐾)β€˜π‘ƒ) = {𝑓 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘¦ ∈ 𝐾 ((π‘“β€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝑓 β€œ π‘₯) βŠ† 𝑦))})
16153impa 1111 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) β†’ ((𝐽 CnP 𝐾)β€˜π‘ƒ) = {𝑓 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘¦ ∈ 𝐾 ((π‘“β€˜π‘ƒ) ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐽 (𝑃 ∈ π‘₯ ∧ (𝑓 β€œ π‘₯) βŠ† 𝑦))})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  βˆƒwrex 3074  {crab 3410   βŠ† wss 3915   ↦ cmpt 5193   β€œ cima 5641  β€˜cfv 6501  (class class class)co 7362   ↑m cmap 8772  TopOnctopon 22275   CnP ccnp 22592
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 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-top 22259  df-topon 22276  df-cnp 22595
This theorem is referenced by:  iscnp  22604
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