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Theorem cnpval 12294
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 12291 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 CnP 𝐾) = (𝑣𝑋 ↦ {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑣) ∈ 𝑦 → ∃𝑥𝐽 (𝑣𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))}))
21fveq1d 5391 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐽 CnP 𝐾)‘𝑃) = ((𝑣𝑋 ↦ {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑣) ∈ 𝑦 → ∃𝑥𝐽 (𝑣𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))})‘𝑃))
32adantr 274 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑃𝑋) → ((𝐽 CnP 𝐾)‘𝑃) = ((𝑣𝑋 ↦ {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑣) ∈ 𝑦 → ∃𝑥𝐽 (𝑣𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))})‘𝑃))
4 eqid 2117 . . . 4 (𝑣𝑋 ↦ {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑣) ∈ 𝑦 → ∃𝑥𝐽 (𝑣𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))}) = (𝑣𝑋 ↦ {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑣) ∈ 𝑦 → ∃𝑥𝐽 (𝑣𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))})
5 fveq2 5389 . . . . . . . 8 (𝑣 = 𝑃 → (𝑓𝑣) = (𝑓𝑃))
65eleq1d 2186 . . . . . . 7 (𝑣 = 𝑃 → ((𝑓𝑣) ∈ 𝑦 ↔ (𝑓𝑃) ∈ 𝑦))
7 eleq1 2180 . . . . . . . . 9 (𝑣 = 𝑃 → (𝑣𝑥𝑃𝑥))
87anbi1d 460 . . . . . . . 8 (𝑣 = 𝑃 → ((𝑣𝑥 ∧ (𝑓𝑥) ⊆ 𝑦) ↔ (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦)))
98rexbidv 2415 . . . . . . 7 (𝑣 = 𝑃 → (∃𝑥𝐽 (𝑣𝑥 ∧ (𝑓𝑥) ⊆ 𝑦) ↔ ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦)))
106, 9imbi12d 233 . . . . . 6 (𝑣 = 𝑃 → (((𝑓𝑣) ∈ 𝑦 → ∃𝑥𝐽 (𝑣𝑥 ∧ (𝑓𝑥) ⊆ 𝑦)) ↔ ((𝑓𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))))
1110ralbidv 2414 . . . . 5 (𝑣 = 𝑃 → (∀𝑦𝐾 ((𝑓𝑣) ∈ 𝑦 → ∃𝑥𝐽 (𝑣𝑥 ∧ (𝑓𝑥) ⊆ 𝑦)) ↔ ∀𝑦𝐾 ((𝑓𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))))
1211rabbidv 2649 . . . 4 (𝑣 = 𝑃 → {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑣) ∈ 𝑦 → ∃𝑥𝐽 (𝑣𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))} = {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))})
13 simpr 109 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑃𝑋) → 𝑃𝑋)
14 fnmap 6517 . . . . . 6 𝑚 Fn (V × V)
15 toponmax 12119 . . . . . . . 8 (𝐾 ∈ (TopOn‘𝑌) → 𝑌𝐾)
1615elexd 2673 . . . . . . 7 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 ∈ V)
1716ad2antlr 480 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑃𝑋) → 𝑌 ∈ V)
18 toponmax 12119 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
1918elexd 2673 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ V)
2019ad2antrr 479 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑃𝑋) → 𝑋 ∈ V)
21 fnovex 5772 . . . . . 6 (( ↑𝑚 Fn (V × V) ∧ 𝑌 ∈ V ∧ 𝑋 ∈ V) → (𝑌𝑚 𝑋) ∈ V)
2214, 17, 20, 21mp3an2i 1305 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑃𝑋) → (𝑌𝑚 𝑋) ∈ V)
23 rabexg 4041 . . . . 5 ((𝑌𝑚 𝑋) ∈ V → {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))} ∈ V)
2422, 23syl 14 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑃𝑋) → {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))} ∈ V)
254, 12, 13, 24fvmptd3 5482 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑃𝑋) → ((𝑣𝑋 ↦ {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑣) ∈ 𝑦 → ∃𝑥𝐽 (𝑣𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))})‘𝑃) = {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))})
263, 25eqtrd 2150 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑃𝑋) → ((𝐽 CnP 𝐾)‘𝑃) = {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))})
27263impa 1161 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 set class
Syntax hints:  wi 4  wa 103  w3a 947   = wceq 1316  wcel 1465  wral 2393  wrex 2394  {crab 2397  Vcvv 2660  wss 3041  cmpt 3959   × cxp 4507  cima 4512   Fn wfn 5088  cfv 5093  (class class class)co 5742  𝑚 cmap 6510  TopOnctopon 12104   CnP ccnp 12282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-map 6512  df-top 12092  df-topon 12105  df-cnp 12285
This theorem is referenced by:  iscnp  12295
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