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Mirrors > Home > MPE Home > Th. List > cnnei | Structured version Visualization version GIF version |
Description: Continuity in terms of neighborhoods. (Contributed by Thierry Arnoux, 3-Jan-2018.) |
Ref | Expression |
---|---|
cnnei.x | ⊢ 𝑋 = ∪ 𝐽 |
cnnei.y | ⊢ 𝑌 = ∪ 𝐾 |
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)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnnei.x | . . . . . 6 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | toptopon 22939 | . . . . 5 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) |
3 | cnnei.y | . . . . . 6 ⊢ 𝑌 = ∪ 𝐾 | |
4 | 3 | toptopon 22939 | . . . . 5 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌)) |
5 | 2, 4 | anbi12i 628 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ↔ (𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌))) |
6 | cncnp 23304 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑝 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑝)))) | |
7 | 6 | baibd 539 | . . . 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 23287 | . . . . . . . 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 539 | . . . . . 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 652 | . . . . 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 3174 | . . 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 279 | . 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)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 ⊆ wss 3963 {csn 4631 ∪ cuni 4912 “ cima 5692 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 Topctop 22915 TopOnctopon 22932 neicnei 23121 Cn ccn 23248 CnP ccnp 23249 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-map 8867 df-topgen 17490 df-top 22916 df-topon 22933 df-ntr 23044 df-nei 23122 df-cn 23251 df-cnp 23252 |
This theorem is referenced by: cnextcn 24091 cnextfres1 24092 |
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