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Theorem iscnp2 23187
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 4333 . . . . . . 7 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → ¬ ((𝐽 CnP 𝐾)‘𝑃) = ∅)
2 df-ov 7422 . . . . . . . . . 10 (𝐽 CnP 𝐾) = ( CnP ‘⟨𝐽, 𝐾⟩)
3 ndmfv 6931 . . . . . . . . . 10 (¬ ⟨𝐽, 𝐾⟩ ∈ dom CnP → ( CnP ‘⟨𝐽, 𝐾⟩) = ∅)
42, 3eqtrid 2777 . . . . . . . . 9 (¬ ⟨𝐽, 𝐾⟩ ∈ dom CnP → (𝐽 CnP 𝐾) = ∅)
54fveq1d 6898 . . . . . . . 8 (¬ ⟨𝐽, 𝐾⟩ ∈ dom CnP → ((𝐽 CnP 𝐾)‘𝑃) = (∅‘𝑃))
6 0fv 6940 . . . . . . . 8 (∅‘𝑃) = ∅
75, 6eqtrdi 2781 . . . . . . 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 23176 . . . . . . 7 CnP = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ (𝑥 𝑗 ↦ {𝑓 ∈ ( 𝑘m 𝑗) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))}))
10 ovex 7452 . . . . . . . . . . 11 ( 𝑘m 𝑗) ∈ V
11 ssrab2 4073 . . . . . . . . . . 11 {𝑓 ∈ ( 𝑘m 𝑗) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))} ⊆ ( 𝑘m 𝑗)
1210, 11elpwi2 5349 . . . . . . . . . 10 {𝑓 ∈ ( 𝑘m 𝑗) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))} ∈ 𝒫 ( 𝑘m 𝑗)
1312rgenw 3054 . . . . . . . . 9 𝑥 𝑗{𝑓 ∈ ( 𝑘m 𝑗) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))} ∈ 𝒫 ( 𝑘m 𝑗)
14 eqid 2725 . . . . . . . . . 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 7119 . . . . . . . . 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 229 . . . . . . . 8 (𝑥 𝑗 ↦ {𝑓 ∈ ( 𝑘m 𝑗) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))}): 𝑗⟶𝒫 ( 𝑘m 𝑗)
17 vuniex 7745 . . . . . . . 8 𝑗 ∈ V
1810pwex 5380 . . . . . . . 8 𝒫 ( 𝑘m 𝑗) ∈ V
19 fex2 7942 . . . . . . . 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 1457 . . . . . . 7 (𝑥 𝑗 ↦ {𝑓 ∈ ( 𝑘m 𝑗) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))}) ∈ V
219, 20dmmpo 8076 . . . . . 6 dom CnP = (Top × Top)
228, 21eleqtrdi 2835 . . . . 5 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → ⟨𝐽, 𝐾⟩ ∈ (Top × Top))
23 opelxp 5714 . . . . 5 (⟨𝐽, 𝐾⟩ ∈ (Top × Top) ↔ (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
2422, 23sylib 217 . . . 4 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
2524simpld 493 . . 3 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐽 ∈ Top)
2624simprd 494 . . 3 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐾 ∈ Top)
27 elfvdm 6933 . . . 4 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝑃 ∈ dom (𝐽 CnP 𝐾))
28 iscn.1 . . . . . . . . 9 𝑋 = 𝐽
2928toptopon 22863 . . . . . . . 8 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
30 iscn.2 . . . . . . . . 9 𝑌 = 𝐾
3130toptopon 22863 . . . . . . . 8 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
32 cnpfval 23182 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 CnP 𝐾) = (𝑥𝑋 ↦ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))}))
3329, 31, 32syl2anb 596 . . . . . . 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 5908 . . . . 5 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → dom (𝐽 CnP 𝐾) = dom (𝑥𝑋 ↦ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))}))
36 ovex 7452 . . . . . . . 8 (𝑌m 𝑋) ∈ V
3736rabex 5335 . . . . . . 7 {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))} ∈ V
3837rgenw 3054 . . . . . 6 𝑥𝑋 {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))} ∈ V
39 dmmptg 6248 . . . . . 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 2781 . . . 4 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → dom (𝐽 CnP 𝐾) = 𝑋)
4227, 41eleqtrd 2827 . . 3 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝑃𝑋)
4325, 26, 423jca 1125 . 2 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃𝑋))
44 biid 260 . . 3 (𝑃𝑋𝑃𝑋)
45 iscnp 23185 . . 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 820 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 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3050  wrex 3059  {crab 3418  Vcvv 3461  wss 3944  c0 4322  𝒫 cpw 4604  cop 4636   cuni 4909  cmpt 5232   × cxp 5676  dom cdm 5678  cima 5681  wf 6545  cfv 6549  (class class class)co 7419  m cmap 8845  Topctop 22839  TopOnctopon 22856   CnP ccnp 23173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-map 8847  df-top 22840  df-topon 22857  df-cnp 23176
This theorem is referenced by:  cnptop1  23190  cnptop2  23191  cnprcl  23193  cnpf  23195  cnpimaex  23204  cnpnei  23212  cnpco  23215  cnprest  23237  cnprest2  23238
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