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Theorem cnnei 23291
Description: Continuity in terms of neighborhoods. (Contributed by Thierry Arnoux, 3-Jan-2018.)
Hypotheses
Ref Expression
cnnei.x 𝑋 = 𝐽
cnnei.y 𝑌 = 𝐾
Assertion
Ref Expression
cnnei ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑝𝑋𝑤 ∈ ((nei‘𝐾)‘{(𝐹𝑝)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑝})(𝐹𝑣) ⊆ 𝑤))
Distinct variable groups:   𝑣,𝑝,𝑤,𝐹   𝐽,𝑝,𝑣,𝑤   𝐾,𝑝,𝑣,𝑤   𝑋,𝑝,𝑣,𝑤   𝑌,𝑝,𝑣,𝑤

Proof of Theorem cnnei
StepHypRef Expression
1 cnnei.x . . . . . 6 𝑋 = 𝐽
21toptopon 22924 . . . . 5 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
3 cnnei.y . . . . . 6 𝑌 = 𝐾
43toptopon 22924 . . . . 5 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
52, 4anbi12i 628 . . . 4 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ↔ (𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)))
6 cncnp 23289 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑝𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑝))))
76baibd 539 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑝𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑝)))
85, 7sylanb 581 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ 𝐹:𝑋𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑝𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑝)))
95anbi1i 624 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ 𝐹:𝑋𝑌) ↔ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌))
10 iscnp4 23272 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑝𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑝) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑤 ∈ ((nei‘𝐾)‘{(𝐹𝑝)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑝})(𝐹𝑣) ⊆ 𝑤)))
11103expa 1118 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑝𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑝) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑤 ∈ ((nei‘𝐾)‘{(𝐹𝑝)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑝})(𝐹𝑣) ⊆ 𝑤)))
1211baibd 539 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑝𝑋) ∧ 𝐹:𝑋𝑌) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑝) ↔ ∀𝑤 ∈ ((nei‘𝐾)‘{(𝐹𝑝)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑝})(𝐹𝑣) ⊆ 𝑤))
1312an32s 652 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑝𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑝) ↔ ∀𝑤 ∈ ((nei‘𝐾)‘{(𝐹𝑝)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑝})(𝐹𝑣) ⊆ 𝑤))
149, 13sylanb 581 . . . 4 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ 𝐹:𝑋𝑌) ∧ 𝑝𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑝) ↔ ∀𝑤 ∈ ((nei‘𝐾)‘{(𝐹𝑝)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑝})(𝐹𝑣) ⊆ 𝑤))
1514ralbidva 3175 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ 𝐹:𝑋𝑌) → (∀𝑝𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑝) ↔ ∀𝑝𝑋𝑤 ∈ ((nei‘𝐾)‘{(𝐹𝑝)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑝})(𝐹𝑣) ⊆ 𝑤))
168, 15bitrd 279 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ 𝐹:𝑋𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑝𝑋𝑤 ∈ ((nei‘𝐾)‘{(𝐹𝑝)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑝})(𝐹𝑣) ⊆ 𝑤))
17163impa 1109 1 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑝𝑋𝑤 ∈ ((nei‘𝐾)‘{(𝐹𝑝)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑝})(𝐹𝑣) ⊆ 𝑤))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wral 3060  wrex 3069  wss 3950  {csn 4625   cuni 4906  cima 5687  wf 6556  cfv 6560  (class class class)co 7432  Topctop 22900  TopOnctopon 22917  neicnei 23106   Cn ccn 23233   CnP ccnp 23234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-map 8869  df-topgen 17489  df-top 22901  df-topon 22918  df-ntr 23029  df-nei 23107  df-cn 23236  df-cnp 23237
This theorem is referenced by:  cnextcn  24076  cnextfres1  24077
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