MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iscnp2 Structured version   Visualization version   GIF version

Theorem iscnp2 23247
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 Mario Carneiro, 21-Aug-2015.)
Hypotheses
Ref Expression
iscn.1 𝑋 = 𝐽
iscn.2 𝑌 = 𝐾
Assertion
Ref Expression
iscnp2 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃𝑋) ∧ (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 iscnp2
Dummy variables 𝑓 𝑔 𝑗 𝑘 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0i 4340 . . . . . . 7 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → ¬ ((𝐽 CnP 𝐾)‘𝑃) = ∅)
2 df-ov 7434 . . . . . . . . . 10 (𝐽 CnP 𝐾) = ( CnP ‘⟨𝐽, 𝐾⟩)
3 ndmfv 6941 . . . . . . . . . 10 (¬ ⟨𝐽, 𝐾⟩ ∈ dom CnP → ( CnP ‘⟨𝐽, 𝐾⟩) = ∅)
42, 3eqtrid 2789 . . . . . . . . 9 (¬ ⟨𝐽, 𝐾⟩ ∈ dom CnP → (𝐽 CnP 𝐾) = ∅)
54fveq1d 6908 . . . . . . . 8 (¬ ⟨𝐽, 𝐾⟩ ∈ dom CnP → ((𝐽 CnP 𝐾)‘𝑃) = (∅‘𝑃))
6 0fv 6950 . . . . . . . 8 (∅‘𝑃) = ∅
75, 6eqtrdi 2793 . . . . . . 7 (¬ ⟨𝐽, 𝐾⟩ ∈ dom CnP → ((𝐽 CnP 𝐾)‘𝑃) = ∅)
81, 7nsyl2 141 . . . . . 6 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → ⟨𝐽, 𝐾⟩ ∈ dom CnP )
9 df-cnp 23236 . . . . . . 7 CnP = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ (𝑥 𝑗 ↦ {𝑓 ∈ ( 𝑘m 𝑗) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))}))
10 ovex 7464 . . . . . . . . . . 11 ( 𝑘m 𝑗) ∈ V
11 ssrab2 4080 . . . . . . . . . . 11 {𝑓 ∈ ( 𝑘m 𝑗) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))} ⊆ ( 𝑘m 𝑗)
1210, 11elpwi2 5335 . . . . . . . . . 10 {𝑓 ∈ ( 𝑘m 𝑗) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))} ∈ 𝒫 ( 𝑘m 𝑗)
1312rgenw 3065 . . . . . . . . 9 𝑥 𝑗{𝑓 ∈ ( 𝑘m 𝑗) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))} ∈ 𝒫 ( 𝑘m 𝑗)
14 eqid 2737 . . . . . . . . . 10 (𝑥 𝑗 ↦ {𝑓 ∈ ( 𝑘m 𝑗) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))}) = (𝑥 𝑗 ↦ {𝑓 ∈ ( 𝑘m 𝑗) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))})
1514fmpt 7130 . . . . . . . . 9 (∀𝑥 𝑗{𝑓 ∈ ( 𝑘m 𝑗) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))} ∈ 𝒫 ( 𝑘m 𝑗) ↔ (𝑥 𝑗 ↦ {𝑓 ∈ ( 𝑘m 𝑗) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))}): 𝑗⟶𝒫 ( 𝑘m 𝑗))
1613, 15mpbi 230 . . . . . . . 8 (𝑥 𝑗 ↦ {𝑓 ∈ ( 𝑘m 𝑗) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))}): 𝑗⟶𝒫 ( 𝑘m 𝑗)
17 vuniex 7759 . . . . . . . 8 𝑗 ∈ V
1810pwex 5380 . . . . . . . 8 𝒫 ( 𝑘m 𝑗) ∈ V
19 fex2 7958 . . . . . . . 8 (((𝑥 𝑗 ↦ {𝑓 ∈ ( 𝑘m 𝑗) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))}): 𝑗⟶𝒫 ( 𝑘m 𝑗) ∧ 𝑗 ∈ V ∧ 𝒫 ( 𝑘m 𝑗) ∈ V) → (𝑥 𝑗 ↦ {𝑓 ∈ ( 𝑘m 𝑗) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))}) ∈ V)
2016, 17, 18, 19mp3an 1463 . . . . . . 7 (𝑥 𝑗 ↦ {𝑓 ∈ ( 𝑘m 𝑗) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))}) ∈ V
219, 20dmmpo 8096 . . . . . 6 dom CnP = (Top × Top)
228, 21eleqtrdi 2851 . . . . 5 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → ⟨𝐽, 𝐾⟩ ∈ (Top × Top))
23 opelxp 5721 . . . . 5 (⟨𝐽, 𝐾⟩ ∈ (Top × Top) ↔ (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
2422, 23sylib 218 . . . 4 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
2524simpld 494 . . 3 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐽 ∈ Top)
2624simprd 495 . . 3 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐾 ∈ Top)
27 elfvdm 6943 . . . 4 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝑃 ∈ dom (𝐽 CnP 𝐾))
28 iscn.1 . . . . . . . . 9 𝑋 = 𝐽
2928toptopon 22923 . . . . . . . 8 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
30 iscn.2 . . . . . . . . 9 𝑌 = 𝐾
3130toptopon 22923 . . . . . . . 8 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
32 cnpfval 23242 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 CnP 𝐾) = (𝑥𝑋 ↦ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))}))
3329, 31, 32syl2anb 598 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 CnP 𝐾) = (𝑥𝑋 ↦ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))}))
3424, 33syl 17 . . . . . 6 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → (𝐽 CnP 𝐾) = (𝑥𝑋 ↦ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))}))
3534dmeqd 5916 . . . . 5 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → dom (𝐽 CnP 𝐾) = dom (𝑥𝑋 ↦ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))}))
36 ovex 7464 . . . . . . . 8 (𝑌m 𝑋) ∈ V
3736rabex 5339 . . . . . . 7 {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))} ∈ V
3837rgenw 3065 . . . . . 6 𝑥𝑋 {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))} ∈ V
39 dmmptg 6262 . . . . . 6 (∀𝑥𝑋 {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))} ∈ V → dom (𝑥𝑋 ↦ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))}) = 𝑋)
4038, 39ax-mp 5 . . . . 5 dom (𝑥𝑋 ↦ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))}) = 𝑋
4135, 40eqtrdi 2793 . . . 4 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → dom (𝐽 CnP 𝐾) = 𝑋)
4227, 41eleqtrd 2843 . . 3 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝑃𝑋)
4325, 26, 423jca 1129 . 2 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃𝑋))
44 biid 261 . . 3 (𝑃𝑋𝑃𝑋)
45 iscnp 23245 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))))
4629, 31, 44, 45syl3anb 1162 . 2 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))))
4743, 46biadanii 822 1 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃𝑋) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  {crab 3436  Vcvv 3480  wss 3951  c0 4333  𝒫 cpw 4600  cop 4632   cuni 4907  cmpt 5225   × cxp 5683  dom cdm 5685  cima 5688  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866  Topctop 22899  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-csb 3900  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-iun 4993  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-1st 8014  df-2nd 8015  df-map 8868  df-top 22900  df-topon 22917  df-cnp 23236
This theorem is referenced by:  cnptop1  23250  cnptop2  23251  cnprcl  23253  cnpf  23255  cnpimaex  23264  cnpnei  23272  cnpco  23275  cnprest  23297  cnprest2  23298
  Copyright terms: Public domain W3C validator