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Mirrors > Home > MPE Home > Th. List > xkotf | Structured version Visualization version GIF version |
Description: Functionality of function π. (Contributed by Mario Carneiro, 19-Mar-2015.) |
Ref | Expression |
---|---|
xkoval.x | β’ π = βͺ π |
xkoval.k | β’ πΎ = {π₯ β π« π β£ (π βΎt π₯) β Comp} |
xkoval.t | β’ π = (π β πΎ, π£ β π β¦ {π β (π Cn π) β£ (π β π) β π£}) |
Ref | Expression |
---|---|
xkotf | β’ π:(πΎ Γ π)βΆπ« (π Cn π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 7435 | . . . 4 β’ (π Cn π) β V | |
2 | ssrab2 4070 | . . . 4 β’ {π β (π Cn π) β£ (π β π) β π£} β (π Cn π) | |
3 | 1, 2 | elpwi2 5337 | . . 3 β’ {π β (π Cn π) β£ (π β π) β π£} β π« (π Cn π) |
4 | 3 | rgen2w 3058 | . 2 β’ βπ β πΎ βπ£ β π {π β (π Cn π) β£ (π β π) β π£} β π« (π Cn π) |
5 | xkoval.t | . . 3 β’ π = (π β πΎ, π£ β π β¦ {π β (π Cn π) β£ (π β π) β π£}) | |
6 | 5 | fmpo 8048 | . 2 β’ (βπ β πΎ βπ£ β π {π β (π Cn π) β£ (π β π) β π£} β π« (π Cn π) β π:(πΎ Γ π)βΆπ« (π Cn π)) |
7 | 4, 6 | mpbi 229 | 1 β’ π:(πΎ Γ π)βΆπ« (π Cn π) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 β wcel 2098 βwral 3053 {crab 3424 Vcvv 3466 β wss 3941 π« cpw 4595 βͺ cuni 4900 Γ cxp 5665 β cima 5670 βΆwf 6530 (class class class)co 7402 β cmpo 7404 βΎt crest 17371 Cn ccn 23072 Compccmp 23234 |
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-sep 5290 ax-nul 5297 ax-pr 5418 ax-un 7719 |
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-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-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-1st 7969 df-2nd 7970 |
This theorem is referenced by: xkoopn 23437 xkouni 23447 xkoccn 23467 xkoco1cn 23505 xkoco2cn 23506 xkococn 23508 xkoinjcn 23535 |
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