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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 6862 | . . . 4 β’ (π β ((neiβπ½)βπ) β π β dom (neiβπ½)) | |
2 | 1 | adantl 482 | . . 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 22357 | . . . . . 6 β’ (π½ β Top β (neiβπ½) Fn π« π) |
5 | 4 | fndmd 6590 | . . . . 5 β’ (π½ β Top β dom (neiβπ½) = π« π) |
6 | 5 | eleq2d 2822 | . . . 4 β’ (π½ β Top β (π β dom (neiβπ½) β π β π« π)) |
7 | 6 | adantr 481 | . . 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 4556 | 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 396 = wceq 1540 β wcel 2105 β wss 3898 π« cpw 4547 βͺ cuni 4852 dom cdm 5620 βcfv 6479 Topctop 22148 neicnei 22354 |
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 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-top 22149 df-nei 22355 |
This theorem is referenced by: neii1 22363 neii2 22365 neiss 22366 ssnei2 22373 topssnei 22381 innei 22382 neitx 22864 cvmlift2lem12 33575 neiin 34617 cnneiima 46561 |
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