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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 4041 | . . . . 5 β’ {π¦ β π½ β£ π₯ β π¦} β π½ | |
2 | elpw2g 5305 | . . . . 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 258 | . . . 4 β’ (π½ β π β {π¦ β π½ β£ π₯ β π¦} β π« π½) |
4 | 3 | adantr 482 | . . 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 7066 | . 2 β’ (π½ β π β πΉ:πβΆπ« π½) |
7 | 6 | ffnd 6673 | 1 β’ (π½ β π β πΉ Fn π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 {crab 3406 β wss 3914 π« cpw 4564 β¦ cmpt 5192 Fn wfn 6495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-fun 6502 df-fn 6503 df-f 6504 |
This theorem is referenced by: kqtopon 23101 kqid 23102 ist0-4 23103 kqfvima 23104 kqsat 23105 kqdisj 23106 kqcldsat 23107 kqopn 23108 kqcld 23109 kqt0lem 23110 isr0 23111 r0cld 23112 regr1lem2 23114 kqreglem1 23115 kqreglem2 23116 kqnrmlem1 23117 kqnrmlem2 23118 |
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