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| Mirrors > Home > MPE Home > Th. List > neiss2 | Structured version Visualization version GIF version | ||
| Description: A set with a neighborhood is a subset of the base set of a topology. (This theorem depends on a function's value being empty outside of its domain, but it will make later theorems simpler to state.) (Contributed by NM, 12-Feb-2007.) |
| Ref | Expression |
|---|---|
| neifval.1 | β’ π = βͺ π½ |
| Ref | Expression |
|---|---|
| neiss2 | β’ ((π½ β Top β§ π β ((neiβπ½)βπ)) β π β π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elfvdm 6934 | . . . 4 β’ (π β ((neiβπ½)βπ) β π β dom (neiβπ½)) | |
| 2 | 1 | adantl 481 | . . 3 β’ ((π½ β Top β§ π β ((neiβπ½)βπ)) β π β dom (neiβπ½)) |
| 3 | neifval.1 | . . . . . . 7 β’ π = βͺ π½ | |
| 4 | 3 | neif 23017 | . . . . . 6 β’ (π½ β Top β (neiβπ½) Fn π« π) |
| 5 | 4 | fndmd 6659 | . . . . 5 β’ (π½ β Top β dom (neiβπ½) = π« π) |
| 6 | 5 | eleq2d 2815 | . . . 4 β’ (π½ β Top β (π β dom (neiβπ½) β π β π« π)) |
| 7 | 6 | adantr 480 | . . 3 β’ ((π½ β Top β§ π β ((neiβπ½)βπ)) β (π β dom (neiβπ½) β π β π« π)) |
| 8 | 2, 7 | mpbid 231 | . 2 β’ ((π½ β Top β§ π β ((neiβπ½)βπ)) β π β π« π) |
| 9 | 8 | elpwid 4612 | 1 β’ ((π½ β Top β§ π β ((neiβπ½)βπ)) β π β π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1534 β wcel 2099 β wss 3947 π« cpw 4603 βͺ cuni 4908 dom cdm 5678 βcfv 6548 Topctop 22808 neicnei 23014 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-top 22809 df-nei 23015 |
| This theorem is referenced by: neii1 23023 neii2 23025 neiss 23026 ssnei2 23033 topssnei 23041 innei 23042 neitx 23524 cvmlift2lem12 34924 neiin 35816 cnneiima 47935 |
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