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Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrneineine0lem | Structured version Visualization version GIF version |
Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, πΉ, then conditions equal to claiming that for every point, at least one (pseudo-)neighborbood exists hold equally. (Contributed by RP, 29-May-2021.) |
Ref | Expression |
---|---|
ntrnei.o | β’ π = (π β V, π β V β¦ (π β (π« π βm π) β¦ (π β π β¦ {π β π β£ π β (πβπ)}))) |
ntrnei.f | β’ πΉ = (π« π΅ππ΅) |
ntrnei.r | β’ (π β πΌπΉπ) |
ntrnei.x | β’ (π β π β π΅) |
Ref | Expression |
---|---|
ntrneineine0lem | β’ (π β (βπ β π« π΅π β (πΌβπ ) β (πβπ) β β )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ntrnei.o | . . . 4 β’ π = (π β V, π β V β¦ (π β (π« π βm π) β¦ (π β π β¦ {π β π β£ π β (πβπ)}))) | |
2 | ntrnei.f | . . . 4 β’ πΉ = (π« π΅ππ΅) | |
3 | ntrnei.r | . . . . 5 β’ (π β πΌπΉπ) | |
4 | 3 | adantr 481 | . . . 4 β’ ((π β§ π β π« π΅) β πΌπΉπ) |
5 | ntrnei.x | . . . . 5 β’ (π β π β π΅) | |
6 | 5 | adantr 481 | . . . 4 β’ ((π β§ π β π« π΅) β π β π΅) |
7 | simpr 485 | . . . 4 β’ ((π β§ π β π« π΅) β π β π« π΅) | |
8 | 1, 2, 4, 6, 7 | ntrneiel 42508 | . . 3 β’ ((π β§ π β π« π΅) β (π β (πΌβπ ) β π β (πβπ))) |
9 | 8 | rexbidva 3175 | . 2 β’ (π β (βπ β π« π΅π β (πΌβπ ) β βπ β π« π΅π β (πβπ))) |
10 | 1, 2, 3 | ntrneinex 42504 | . . . . . . . . . 10 β’ (π β π β (π« π« π΅ βm π΅)) |
11 | elmapi 8809 | . . . . . . . . . 10 β’ (π β (π« π« π΅ βm π΅) β π:π΅βΆπ« π« π΅) | |
12 | 10, 11 | syl 17 | . . . . . . . . 9 β’ (π β π:π΅βΆπ« π« π΅) |
13 | 12, 5 | ffvelcdmd 7056 | . . . . . . . 8 β’ (π β (πβπ) β π« π« π΅) |
14 | 13 | elpwid 4589 | . . . . . . 7 β’ (π β (πβπ) β π« π΅) |
15 | 14 | sseld 3961 | . . . . . 6 β’ (π β (π β (πβπ) β π β π« π΅)) |
16 | 15 | pm4.71rd 563 | . . . . 5 β’ (π β (π β (πβπ) β (π β π« π΅ β§ π β (πβπ)))) |
17 | 16 | exbidv 1924 | . . . 4 β’ (π β (βπ π β (πβπ) β βπ (π β π« π΅ β§ π β (πβπ)))) |
18 | 17 | bicomd 222 | . . 3 β’ (π β (βπ (π β π« π΅ β§ π β (πβπ)) β βπ π β (πβπ))) |
19 | df-rex 3070 | . . 3 β’ (βπ β π« π΅π β (πβπ) β βπ (π β π« π΅ β§ π β (πβπ))) | |
20 | n0 4326 | . . 3 β’ ((πβπ) β β β βπ π β (πβπ)) | |
21 | 18, 19, 20 | 3bitr4g 313 | . 2 β’ (π β (βπ β π« π΅π β (πβπ) β (πβπ) β β )) |
22 | 9, 21 | bitrd 278 | 1 β’ (π β (βπ β π« π΅π β (πΌβπ ) β (πβπ) β β )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 396 = wceq 1541 βwex 1781 β wcel 2106 β wne 2939 βwrex 3069 {crab 3418 Vcvv 3459 β c0 4302 π« cpw 4580 class class class wbr 5125 β¦ cmpt 5208 βΆwf 6512 βcfv 6516 (class class class)co 7377 β cmpo 7379 βm cmap 8787 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5262 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-iun 4976 df-br 5126 df-opab 5188 df-mpt 5209 df-id 5551 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-ov 7380 df-oprab 7381 df-mpo 7382 df-1st 7941 df-2nd 7942 df-map 8789 |
This theorem is referenced by: ntrneineine0 42514 |
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