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Mirrors > Home > ILE Home > Th. List > cnnei | 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 12810 | . . . . 5 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) |
3 | cnnei.y | . . . . . 6 ⊢ 𝑌 = ∪ 𝐾 | |
4 | 3 | toptopon 12810 | . . . . 5 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌)) |
5 | 2, 4 | anbi12i 457 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ↔ (𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌))) |
6 | cncnp 13024 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑝 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑝)))) | |
7 | 6 | baibd 918 | . . . 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 282 | . . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ 𝐹:𝑋⟶𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑝 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑝))) |
9 | 5 | anbi1i 455 | . . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ 𝐹:𝑋⟶𝑌) ↔ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌)) |
10 | iscnp4 13012 | . . . . . . . 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 1198 | . . . . . . 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 918 | . . . . . 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 563 | . . . . 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 282 | . . . 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 2466 | . . 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 187 | . 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 1189 | 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 set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 973 = wceq 1348 ∈ wcel 2141 ∀wral 2448 ∃wrex 2449 ⊆ wss 3121 {csn 3583 ∪ cuni 3796 “ cima 4614 ⟶wf 5194 ‘cfv 5198 (class class class)co 5853 Topctop 12789 TopOnctopon 12802 neicnei 12932 Cn ccn 12979 CnP ccnp 12980 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-map 6628 df-topgen 12600 df-top 12790 df-topon 12803 df-ntr 12890 df-nei 12933 df-cn 12982 df-cnp 12983 |
This theorem is referenced by: (None) |
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