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Theorem cnpval 21563
 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 21561 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 CnP 𝐾) = (𝑣𝑋 ↦ {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑣) ∈ 𝑦 → ∃𝑥𝐽 (𝑣𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))}))
21fveq1d 6506 . . 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 6504 . . . . . . . 8 (𝑣 = 𝑃 → (𝑓𝑣) = (𝑓𝑃))
43eleq1d 2852 . . . . . . 7 (𝑣 = 𝑃 → ((𝑓𝑣) ∈ 𝑦 ↔ (𝑓𝑃) ∈ 𝑦))
5 eleq1 2855 . . . . . . . . 9 (𝑣 = 𝑃 → (𝑣𝑥𝑃𝑥))
65anbi1d 621 . . . . . . . 8 (𝑣 = 𝑃 → ((𝑣𝑥 ∧ (𝑓𝑥) ⊆ 𝑦) ↔ (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦)))
76rexbidv 3244 . . . . . . 7 (𝑣 = 𝑃 → (∃𝑥𝐽 (𝑣𝑥 ∧ (𝑓𝑥) ⊆ 𝑦) ↔ ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦)))
84, 7imbi12d 337 . . . . . 6 (𝑣 = 𝑃 → (((𝑓𝑣) ∈ 𝑦 → ∃𝑥𝐽 (𝑣𝑥 ∧ (𝑓𝑥) ⊆ 𝑦)) ↔ ((𝑓𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))))
98ralbidv 3149 . . . . 5 (𝑣 = 𝑃 → (∀𝑦𝐾 ((𝑓𝑣) ∈ 𝑦 → ∃𝑥𝐽 (𝑣𝑥 ∧ (𝑓𝑥) ⊆ 𝑦)) ↔ ∀𝑦𝐾 ((𝑓𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))))
109rabbidv 3405 . . . 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 2780 . . . 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 7014 . . . . 5 (𝑌𝑚 𝑋) ∈ V
1312rabex 5095 . . . 4 {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))} ∈ V
1410, 11, 13fvmpt 6601 . . 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 2836 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑃𝑋) → ((𝐽 CnP 𝐾)‘𝑃) = {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))})
16153impa 1091 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 387   ∧ w3a 1069   = wceq 1508   ∈ wcel 2051  ∀wral 3090  ∃wrex 3091  {crab 3094   ⊆ wss 3831   ↦ cmpt 5013   “ cima 5414  ‘cfv 6193  (class class class)co 6982   ↑𝑚 cmap 8212  TopOnctopon 21237   CnP ccnp 21552 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2752  ax-sep 5064  ax-nul 5071  ax-pow 5123  ax-pr 5190  ax-un 7285 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2551  df-eu 2589  df-clab 2761  df-cleq 2773  df-clel 2848  df-nfc 2920  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3419  df-sbc 3684  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4182  df-if 4354  df-pw 4427  df-sn 4445  df-pr 4447  df-op 4451  df-uni 4718  df-br 4935  df-opab 4997  df-mpt 5014  df-id 5316  df-xp 5417  df-rel 5418  df-cnv 5419  df-co 5420  df-dm 5421  df-rn 5422  df-res 5423  df-ima 5424  df-iota 6157  df-fun 6195  df-fn 6196  df-f 6197  df-fv 6201  df-ov 6985  df-oprab 6986  df-mpo 6987  df-top 21221  df-topon 21238  df-cnp 21555 This theorem is referenced by:  iscnp  21564
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