Theorem cnnei 22516

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:𝑋⟶𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑝 ∈ 𝑋 ∀𝑤 ∈ ((nei‘𝐾)‘{(𝐹‘𝑝)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑝})(𝐹 “ 𝑣) ⊆ 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

Step	Hyp	Ref	Expression
1		cnnei.x	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
2	1	toptopon 22149	. . . . 5 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
3		cnnei.y	. . . . . 6 ⊢ 𝑌 = ∪ 𝐾
4	3	toptopon 22149	. . . . 5 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
5	2, 4	anbi12i 627	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ↔ (𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)))
6		cncnp 22514	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑝 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑝))))
7	6	baibd 540	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑝 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑝)))
8	5, 7	sylanb 581	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ 𝐹:𝑋⟶𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑝 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑝)))
9	5	anbi1i 624	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ 𝐹:𝑋⟶𝑌) ↔ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌))
10		iscnp4 22497	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑝 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑝) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑤 ∈ ((nei‘𝐾)‘{(𝐹‘𝑝)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑝})(𝐹 “ 𝑣) ⊆ 𝑤)))
11	10	3expa 1117	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑝 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑝) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑤 ∈ ((nei‘𝐾)‘{(𝐹‘𝑝)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑝})(𝐹 “ 𝑣) ⊆ 𝑤)))
12	11	baibd 540	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑝 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑝) ↔ ∀𝑤 ∈ ((nei‘𝐾)‘{(𝐹‘𝑝)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑝})(𝐹 “ 𝑣) ⊆ 𝑤))
13	12	an32s 649	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑝 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑝) ↔ ∀𝑤 ∈ ((nei‘𝐾)‘{(𝐹‘𝑝)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑝})(𝐹 “ 𝑣) ⊆ 𝑤))
14	9, 13	sylanb 581	. . . 4 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑝 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑝) ↔ ∀𝑤 ∈ ((nei‘𝐾)‘{(𝐹‘𝑝)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑝})(𝐹 “ 𝑣) ⊆ 𝑤))
15	14	ralbidva 3169	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑝 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑝) ↔ ∀𝑝 ∈ 𝑋 ∀𝑤 ∈ ((nei‘𝐾)‘{(𝐹‘𝑝)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑝})(𝐹 “ 𝑣) ⊆ 𝑤))
16	8, 15	bitrd 278	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ 𝐹:𝑋⟶𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑝 ∈ 𝑋 ∀𝑤 ∈ ((nei‘𝐾)‘{(𝐹‘𝑝)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑝})(𝐹 “ 𝑣) ⊆ 𝑤))
17	16	3impa 1109	1 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑝 ∈ 𝑋 ∀𝑤 ∈ ((nei‘𝐾)‘{(𝐹‘𝑝)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑝})(𝐹 “ 𝑣) ⊆ 𝑤))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ∀wral 3062  ∃wrex 3071   ⊆ wss 3897  {csn 4571  ∪ cuni 4850   “ cima 5611  ⟶wf 6462  ‘cfv 6466  (class class class)co 7317  Topctop 22125  TopOnctopon 22142  neicnei 22331   Cn ccn 22458   CnP ccnp 22459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5224  ax-sep 5238  ax-nul 5245  ax-pow 5303  ax-pr 5367  ax-un 7630

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4268  df-if 4472  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-iun 4939  df-br 5088  df-opab 5150  df-mpt 5171  df-id 5507  df-xp 5614  df-rel 5615  df-cnv 5616  df-co 5617  df-dm 5618  df-rn 5619  df-res 5620  df-ima 5621  df-iota 6418  df-fun 6468  df-fn 6469  df-f 6470  df-f1 6471  df-fo 6472  df-f1o 6473  df-fv 6474  df-ov 7320  df-oprab 7321  df-mpo 7322  df-1st 7878  df-2nd 7879  df-map 8667  df-topgen 17231  df-top 22126  df-topon 22143  df-ntr 22254  df-nei 22332  df-cn 22461  df-cnp 22462

This theorem is referenced by:  cnextcn  23301  cnextfres1  23302