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Theorem cnpval 21839
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 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) → ((𝐽 CnP 𝐾)‘𝑃) = {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))})
Distinct variable groups:   𝑥,𝑓,𝑦,𝐽   𝑓,𝐾,𝑥,𝑦   𝑓,𝑋,𝑥,𝑦   𝑃,𝑓,𝑥,𝑦   𝑓,𝑌,𝑥,𝑦

Proof of Theorem cnpval
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 cnpfval 21837 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 CnP 𝐾) = (𝑣𝑋 ↦ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑣) ∈ 𝑦 → ∃𝑥𝐽 (𝑣𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))}))
21fveq1d 6654 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐽 CnP 𝐾)‘𝑃) = ((𝑣𝑋 ↦ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑣) ∈ 𝑦 → ∃𝑥𝐽 (𝑣𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))})‘𝑃))
3 fveq2 6652 . . . . . . . 8 (𝑣 = 𝑃 → (𝑓𝑣) = (𝑓𝑃))
43eleq1d 2898 . . . . . . 7 (𝑣 = 𝑃 → ((𝑓𝑣) ∈ 𝑦 ↔ (𝑓𝑃) ∈ 𝑦))
5 eleq1 2901 . . . . . . . . 9 (𝑣 = 𝑃 → (𝑣𝑥𝑃𝑥))
65anbi1d 632 . . . . . . . 8 (𝑣 = 𝑃 → ((𝑣𝑥 ∧ (𝑓𝑥) ⊆ 𝑦) ↔ (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦)))
76rexbidv 3283 . . . . . . 7 (𝑣 = 𝑃 → (∃𝑥𝐽 (𝑣𝑥 ∧ (𝑓𝑥) ⊆ 𝑦) ↔ ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦)))
84, 7imbi12d 348 . . . . . 6 (𝑣 = 𝑃 → (((𝑓𝑣) ∈ 𝑦 → ∃𝑥𝐽 (𝑣𝑥 ∧ (𝑓𝑥) ⊆ 𝑦)) ↔ ((𝑓𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))))
98ralbidv 3187 . . . . 5 (𝑣 = 𝑃 → (∀𝑦𝐾 ((𝑓𝑣) ∈ 𝑦 → ∃𝑥𝐽 (𝑣𝑥 ∧ (𝑓𝑥) ⊆ 𝑦)) ↔ ∀𝑦𝐾 ((𝑓𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))))
109rabbidv 3455 . . . 4 (𝑣 = 𝑃 → {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑣) ∈ 𝑦 → ∃𝑥𝐽 (𝑣𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))} = {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))})
11 eqid 2822 . . . 4 (𝑣𝑋 ↦ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑣) ∈ 𝑦 → ∃𝑥𝐽 (𝑣𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))}) = (𝑣𝑋 ↦ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑣) ∈ 𝑦 → ∃𝑥𝐽 (𝑣𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))})
12 ovex 7173 . . . . 5 (𝑌m 𝑋) ∈ V
1312rabex 5211 . . . 4 {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))} ∈ V
1410, 11, 13fvmpt 6750 . . 3 (𝑃𝑋 → ((𝑣𝑋 ↦ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑣) ∈ 𝑦 → ∃𝑥𝐽 (𝑣𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))})‘𝑃) = {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))})
152, 14sylan9eq 2877 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑃𝑋) → ((𝐽 CnP 𝐾)‘𝑃) = {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))})
16153impa 1107 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) → ((𝐽 CnP 𝐾)‘𝑃) = {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2114  wral 3130  wrex 3131  {crab 3134  wss 3908  cmpt 5122  cima 5535  cfv 6334  (class class class)co 7140  m cmap 8393  TopOnctopon 21513   CnP ccnp 21828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-top 21497  df-topon 21514  df-cnp 21831
This theorem is referenced by:  iscnp  21840
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