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Theorem cnpval 12426
 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 12423 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 CnP 𝐾) = (𝑣𝑋 ↦ {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑣) ∈ 𝑦 → ∃𝑥𝐽 (𝑣𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))}))
21fveq1d 5432 . . . 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 2140 . . . 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 5430 . . . . . . . 8 (𝑣 = 𝑃 → (𝑓𝑣) = (𝑓𝑃))
65eleq1d 2209 . . . . . . 7 (𝑣 = 𝑃 → ((𝑓𝑣) ∈ 𝑦 ↔ (𝑓𝑃) ∈ 𝑦))
7 eleq1 2203 . . . . . . . . 9 (𝑣 = 𝑃 → (𝑣𝑥𝑃𝑥))
87anbi1d 461 . . . . . . . 8 (𝑣 = 𝑃 → ((𝑣𝑥 ∧ (𝑓𝑥) ⊆ 𝑦) ↔ (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦)))
98rexbidv 2440 . . . . . . 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 2439 . . . . 5 (𝑣 = 𝑃 → (∀𝑦𝐾 ((𝑓𝑣) ∈ 𝑦 → ∃𝑥𝐽 (𝑣𝑥 ∧ (𝑓𝑥) ⊆ 𝑦)) ↔ ∀𝑦𝐾 ((𝑓𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))))
1211rabbidv 2679 . . . 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 6558 . . . . . 6 𝑚 Fn (V × V)
15 toponmax 12251 . . . . . . . 8 (𝐾 ∈ (TopOn‘𝑌) → 𝑌𝐾)
1615elexd 2703 . . . . . . 7 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 ∈ V)
1716ad2antlr 481 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑃𝑋) → 𝑌 ∈ V)
18 toponmax 12251 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
1918elexd 2703 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ V)
2019ad2antrr 480 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑃𝑋) → 𝑋 ∈ V)
21 fnovex 5813 . . . . . 6 (( ↑𝑚 Fn (V × V) ∧ 𝑌 ∈ V ∧ 𝑋 ∈ V) → (𝑌𝑚 𝑋) ∈ V)
2214, 17, 20, 21mp3an2i 1321 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑃𝑋) → (𝑌𝑚 𝑋) ∈ V)
23 rabexg 4080 . . . . 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 5523 . . 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 2173 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑃𝑋) → ((𝐽 CnP 𝐾)‘𝑃) = {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))})
27263impa 1177 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 963   = wceq 1332   ∈ wcel 1481  ∀wral 2417  ∃wrex 2418  {crab 2421  Vcvv 2690   ⊆ wss 3077   ↦ cmpt 3998   × cxp 4546   “ cima 4551   Fn wfn 5127  ‘cfv 5132  (class class class)co 5783   ↑𝑚 cmap 6551  TopOnctopon 12236   CnP ccnp 12414 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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-pow 4107  ax-pr 4140  ax-un 4364  ax-setind 4461 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2692  df-sbc 2915  df-csb 3009  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-iun 3824  df-br 3939  df-opab 3999  df-mpt 4000  df-id 4224  df-xp 4554  df-rel 4555  df-cnv 4556  df-co 4557  df-dm 4558  df-rn 4559  df-res 4560  df-ima 4561  df-iota 5097  df-fun 5134  df-fn 5135  df-f 5136  df-fv 5140  df-ov 5786  df-oprab 5787  df-mpo 5788  df-1st 6047  df-2nd 6048  df-map 6553  df-top 12224  df-topon 12237  df-cnp 12417 This theorem is referenced by:  iscnp  12427
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