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Theorem iscnp 23245
Description: The predicate "the class 𝐹 is a continuous function from topology 𝐽 to topology 𝐾 at point 𝑃". Based on Theorem 7.2(g) of [Munkres] p. 107. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
iscnp ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))))
Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝐾,𝑦   𝑥,𝑋,𝑦   𝑥,𝐹,𝑦   𝑥,𝑃,𝑦   𝑥,𝑌,𝑦

Proof of Theorem iscnp
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 cnpval 23244 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) → ((𝐽 CnP 𝐾)‘𝑃) = {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))})
21eleq2d 2827 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ 𝐹 ∈ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))}))
3 fveq1 6905 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓𝑃) = (𝐹𝑃))
43eleq1d 2826 . . . . . . 7 (𝑓 = 𝐹 → ((𝑓𝑃) ∈ 𝑦 ↔ (𝐹𝑃) ∈ 𝑦))
5 imaeq1 6073 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
65sseq1d 4015 . . . . . . . . 9 (𝑓 = 𝐹 → ((𝑓𝑥) ⊆ 𝑦 ↔ (𝐹𝑥) ⊆ 𝑦))
76anbi2d 630 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦) ↔ (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))
87rexbidv 3179 . . . . . . 7 (𝑓 = 𝐹 → (∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦) ↔ ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))
94, 8imbi12d 344 . . . . . 6 (𝑓 = 𝐹 → (((𝑓𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦)) ↔ ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦))))
109ralbidv 3178 . . . . 5 (𝑓 = 𝐹 → (∀𝑦𝐾 ((𝑓𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦)) ↔ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦))))
1110elrab 3692 . . . 4 (𝐹 ∈ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))} ↔ (𝐹 ∈ (𝑌m 𝑋) ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦))))
12 toponmax 22932 . . . . . 6 (𝐾 ∈ (TopOn‘𝑌) → 𝑌𝐾)
13 toponmax 22932 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
14 elmapg 8879 . . . . . 6 ((𝑌𝐾𝑋𝐽) → (𝐹 ∈ (𝑌m 𝑋) ↔ 𝐹:𝑋𝑌))
1512, 13, 14syl2anr 597 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝑌m 𝑋) ↔ 𝐹:𝑋𝑌))
1615anbi1d 631 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹 ∈ (𝑌m 𝑋) ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦))) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))))
1711, 16bitrid 283 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))} ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))))
18173adant3 1133 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) → (𝐹 ∈ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))} ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))))
192, 18bitrd 279 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  {crab 3436  wss 3951  cima 5688  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866  TopOnctopon 22916   CnP ccnp 23233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-top 22900  df-topon 22917  df-cnp 23236
This theorem is referenced by:  iscnp2  23247  iscnp3  23252  tgcnp  23261  iscnp4  23271  cnconst2  23291  cnpresti  23296  cnprest  23297  cnprest2  23298  1stccnp  23470  cnpflf2  24008  symgtgp  24114  ghmcnp  24123  ellimc2  25912  xrlimcnp  27011  poimir  37660  icccncfext  45902
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