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Theorem cnpval 21021
 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 21019 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 CnP 𝐾) = (𝑣𝑋 ↦ {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑣) ∈ 𝑦 → ∃𝑥𝐽 (𝑣𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))}))
21fveq1d 6180 . . 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 6178 . . . . . . . 8 (𝑣 = 𝑃 → (𝑓𝑣) = (𝑓𝑃))
43eleq1d 2684 . . . . . . 7 (𝑣 = 𝑃 → ((𝑓𝑣) ∈ 𝑦 ↔ (𝑓𝑃) ∈ 𝑦))
5 eleq1 2687 . . . . . . . . 9 (𝑣 = 𝑃 → (𝑣𝑥𝑃𝑥))
65anbi1d 740 . . . . . . . 8 (𝑣 = 𝑃 → ((𝑣𝑥 ∧ (𝑓𝑥) ⊆ 𝑦) ↔ (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦)))
76rexbidv 3048 . . . . . . 7 (𝑣 = 𝑃 → (∃𝑥𝐽 (𝑣𝑥 ∧ (𝑓𝑥) ⊆ 𝑦) ↔ ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦)))
84, 7imbi12d 334 . . . . . 6 (𝑣 = 𝑃 → (((𝑓𝑣) ∈ 𝑦 → ∃𝑥𝐽 (𝑣𝑥 ∧ (𝑓𝑥) ⊆ 𝑦)) ↔ ((𝑓𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))))
98ralbidv 2983 . . . . 5 (𝑣 = 𝑃 → (∀𝑦𝐾 ((𝑓𝑣) ∈ 𝑦 → ∃𝑥𝐽 (𝑣𝑥 ∧ (𝑓𝑥) ⊆ 𝑦)) ↔ ∀𝑦𝐾 ((𝑓𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))))
109rabbidv 3184 . . . 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 2620 . . . 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 6663 . . . . 5 (𝑌𝑚 𝑋) ∈ V
1312rabex 4804 . . . 4 {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))} ∈ V
1410, 11, 13fvmpt 6269 . . 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 2674 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑃𝑋) → ((𝐽 CnP 𝐾)‘𝑃) = {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))})
16153impa 1257 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 384   ∧ w3a 1036   = wceq 1481   ∈ wcel 1988  ∀wral 2909  ∃wrex 2910  {crab 2913   ⊆ wss 3567   ↦ cmpt 4720   “ cima 5107  ‘cfv 5876  (class class class)co 6635   ↑𝑚 cmap 7842  TopOnctopon 20696   CnP ccnp 21010 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-top 20680  df-topon 20697  df-cnp 21013 This theorem is referenced by:  iscnp  21022
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