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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 6919 | . . . 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 22948 | . . . . . 6 β’ (π½ β Top β (neiβπ½) Fn π« π) |
| 5 | 4 | fndmd 6645 | . . . . 5 β’ (π½ β Top β dom (neiβπ½) = π« π) |
| 6 | 5 | eleq2d 2811 | . . . 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 4604 | 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 1533 β wcel 2098 β wss 3941 π« cpw 4595 βͺ cuni 4900 dom cdm 5667 βcfv 6534 Topctop 22739 neicnei 22945 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-top 22740 df-nei 22946 |
| This theorem is referenced by: neii1 22954 neii2 22956 neiss 22957 ssnei2 22964 topssnei 22972 innei 22973 neitx 23455 cvmlift2lem12 34823 neiin 35718 cnneiima 47797 |
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