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Mirrors > Home > MPE Home > Th. List > kqffn | Structured version Visualization version GIF version |
Description: The topological indistinguishability map is a function on the base. (Contributed by Mario Carneiro, 25-Aug-2015.) |
Ref | Expression |
---|---|
kqval.2 | β’ πΉ = (π₯ β π β¦ {π¦ β π½ β£ π₯ β π¦}) |
Ref | Expression |
---|---|
kqffn | β’ (π½ β π β πΉ Fn π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 4077 | . . . . 5 β’ {π¦ β π½ β£ π₯ β π¦} β π½ | |
2 | elpw2g 5344 | . . . . 5 β’ (π½ β π β ({π¦ β π½ β£ π₯ β π¦} β π« π½ β {π¦ β π½ β£ π₯ β π¦} β π½)) | |
3 | 1, 2 | mpbiri 257 | . . . 4 β’ (π½ β π β {π¦ β π½ β£ π₯ β π¦} β π« π½) |
4 | 3 | adantr 481 | . . 3 β’ ((π½ β π β§ π₯ β π) β {π¦ β π½ β£ π₯ β π¦} β π« π½) |
5 | kqval.2 | . . 3 β’ πΉ = (π₯ β π β¦ {π¦ β π½ β£ π₯ β π¦}) | |
6 | 4, 5 | fmptd 7113 | . 2 β’ (π½ β π β πΉ:πβΆπ« π½) |
7 | 6 | ffnd 6718 | 1 β’ (π½ β π β πΉ Fn π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1541 β wcel 2106 {crab 3432 β wss 3948 π« cpw 4602 β¦ cmpt 5231 Fn wfn 6538 |
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 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-fun 6545 df-fn 6546 df-f 6547 |
This theorem is referenced by: kqtopon 23230 kqid 23231 ist0-4 23232 kqfvima 23233 kqsat 23234 kqdisj 23235 kqcldsat 23236 kqopn 23237 kqcld 23238 kqt0lem 23239 isr0 23240 r0cld 23241 regr1lem2 23243 kqreglem1 23244 kqreglem2 23245 kqnrmlem1 23246 kqnrmlem2 23247 |
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