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

Theorem iscnp2 21851
 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 4249 . . . . . . 7 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → ¬ ((𝐽 CnP 𝐾)‘𝑃) = ∅)
2 df-ov 7138 . . . . . . . . . 10 (𝐽 CnP 𝐾) = ( CnP ‘⟨𝐽, 𝐾⟩)
3 ndmfv 6675 . . . . . . . . . 10 (¬ ⟨𝐽, 𝐾⟩ ∈ dom CnP → ( CnP ‘⟨𝐽, 𝐾⟩) = ∅)
42, 3syl5eq 2845 . . . . . . . . 9 (¬ ⟨𝐽, 𝐾⟩ ∈ dom CnP → (𝐽 CnP 𝐾) = ∅)
54fveq1d 6647 . . . . . . . 8 (¬ ⟨𝐽, 𝐾⟩ ∈ dom CnP → ((𝐽 CnP 𝐾)‘𝑃) = (∅‘𝑃))
6 0fv 6684 . . . . . . . 8 (∅‘𝑃) = ∅
75, 6eqtrdi 2849 . . . . . . 7 (¬ ⟨𝐽, 𝐾⟩ ∈ dom CnP → ((𝐽 CnP 𝐾)‘𝑃) = ∅)
81, 7nsyl2 143 . . . . . 6 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → ⟨𝐽, 𝐾⟩ ∈ dom CnP )
9 df-cnp 21840 . . . . . . 7 CnP = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ (𝑥 𝑗 ↦ {𝑓 ∈ ( 𝑘m 𝑗) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))}))
10 ovex 7168 . . . . . . . . . . 11 ( 𝑘m 𝑗) ∈ V
11 ssrab2 4007 . . . . . . . . . . 11 {𝑓 ∈ ( 𝑘m 𝑗) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))} ⊆ ( 𝑘m 𝑗)
1210, 11elpwi2 5213 . . . . . . . . . 10 {𝑓 ∈ ( 𝑘m 𝑗) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))} ∈ 𝒫 ( 𝑘m 𝑗)
1312rgenw 3118 . . . . . . . . 9 𝑥 𝑗{𝑓 ∈ ( 𝑘m 𝑗) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))} ∈ 𝒫 ( 𝑘m 𝑗)
14 eqid 2798 . . . . . . . . . 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 6851 . . . . . . . . 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 233 . . . . . . . 8 (𝑥 𝑗 ↦ {𝑓 ∈ ( 𝑘m 𝑗) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))}): 𝑗⟶𝒫 ( 𝑘m 𝑗)
17 vuniex 7447 . . . . . . . 8 𝑗 ∈ V
1810pwex 5246 . . . . . . . 8 𝒫 ( 𝑘m 𝑗) ∈ V
19 fex2 7622 . . . . . . . 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 1458 . . . . . . 7 (𝑥 𝑗 ↦ {𝑓 ∈ ( 𝑘m 𝑗) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))}) ∈ V
219, 20dmmpo 7753 . . . . . 6 dom CnP = (Top × Top)
228, 21eleqtrdi 2900 . . . . 5 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → ⟨𝐽, 𝐾⟩ ∈ (Top × Top))
23 opelxp 5555 . . . . 5 (⟨𝐽, 𝐾⟩ ∈ (Top × Top) ↔ (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
2422, 23sylib 221 . . . 4 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
2524simpld 498 . . 3 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐽 ∈ Top)
2624simprd 499 . . 3 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐾 ∈ Top)
27 elfvdm 6677 . . . 4 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝑃 ∈ dom (𝐽 CnP 𝐾))
28 iscn.1 . . . . . . . . 9 𝑋 = 𝐽
2928toptopon 21529 . . . . . . . 8 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
30 iscn.2 . . . . . . . . 9 𝑌 = 𝐾
3130toptopon 21529 . . . . . . . 8 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
32 cnpfval 21846 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 CnP 𝐾) = (𝑥𝑋 ↦ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))}))
3329, 31, 32syl2anb 600 . . . . . . 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 5738 . . . . 5 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → dom (𝐽 CnP 𝐾) = dom (𝑥𝑋 ↦ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))}))
36 ovex 7168 . . . . . . . 8 (𝑌m 𝑋) ∈ V
3736rabex 5199 . . . . . . 7 {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))} ∈ V
3837rgenw 3118 . . . . . 6 𝑥𝑋 {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))} ∈ V
39 dmmptg 6063 . . . . . 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 2849 . . . 4 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → dom (𝐽 CnP 𝐾) = 𝑋)
4227, 41eleqtrd 2892 . . 3 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝑃𝑋)
4325, 26, 423jca 1125 . 2 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃𝑋))
44 biid 264 . . 3 (𝑃𝑋𝑃𝑋)
45 iscnp 21849 . . 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 1158 . 2 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))))
4743, 46biadanii 821 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  {crab 3110  Vcvv 3441   ⊆ wss 3881  ∅c0 4243  𝒫 cpw 4497  ⟨cop 4531  ∪ cuni 4800   ↦ cmpt 5110   × cxp 5517  dom cdm 5519   “ cima 5522  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ↑m cmap 8391  Topctop 21505  TopOnctopon 21522   CnP ccnp 21837 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7673  df-2nd 7674  df-map 8393  df-top 21506  df-topon 21523  df-cnp 21840 This theorem is referenced by:  cnptop1  21854  cnptop2  21855  cnprcl  21857  cnpf  21859  cnpimaex  21868  cnpnei  21876  cnpco  21879  cnprest  21901  cnprest2  21902
 Copyright terms: Public domain W3C validator