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Theorem cnnei 21885
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 21520 . . . . 5 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
3 cnnei.y . . . . . 6 𝑌 = 𝐾
43toptopon 21520 . . . . 5 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
52, 4anbi12i 629 . . . 4 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ↔ (𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)))
6 cncnp 21883 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑝𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑝))))
76baibd 543 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑝𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑝)))
85, 7sylanb 584 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ 𝐹:𝑋𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑝𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑝)))
95anbi1i 626 . . . . 5 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ 𝐹:𝑋𝑌) ↔ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌))
10 iscnp4 21866 . . . . . . . 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 1115 . . . . . . 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 543 . . . . . 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 651 . . . . 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 584 . . . 4 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ 𝐹:𝑋𝑌) ∧ 𝑝𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑝) ↔ ∀𝑤 ∈ ((nei‘𝐾)‘{(𝐹𝑝)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑝})(𝐹𝑣) ⊆ 𝑤))
1514ralbidva 3191 . . 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 282 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ 𝐹:𝑋𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑝𝑋𝑤 ∈ ((nei‘𝐾)‘{(𝐹𝑝)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑝})(𝐹𝑣) ⊆ 𝑤))
17163impa 1107 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3133  wrex 3134  wss 3919  {csn 4550   cuni 4825  cima 5546  wf 6340  cfv 6344  (class class class)co 7146  Topctop 21496  TopOnctopon 21513  neicnei 21700   Cn ccn 21827   CnP ccnp 21828
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-ov 7149  df-oprab 7150  df-mpo 7151  df-1st 7681  df-2nd 7682  df-map 8400  df-topgen 16715  df-top 21497  df-topon 21514  df-ntr 21623  df-nei 21701  df-cn 21830  df-cnp 21831
This theorem is referenced by:  cnextcn  22670  cnextfres1  22671
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