![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 7453 | . . . 4 β’ (π Cn π) β V | |
2 | ssrab2 4075 | . . . 4 β’ {π β (π Cn π) β£ (π β π) β π£} β (π Cn π) | |
3 | 1, 2 | elpwi2 5348 | . . 3 β’ {π β (π Cn π) β£ (π β π) β π£} β π« (π Cn π) |
4 | 3 | rgen2w 3063 | . 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 8072 | . 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 1534 β wcel 2099 βwral 3058 {crab 3429 Vcvv 3471 β wss 3947 π« cpw 4603 βͺ cuni 4908 Γ cxp 5676 β cima 5681 βΆwf 6544 (class class class)co 7420 β cmpo 7422 βΎt crest 17402 Cn ccn 23141 Compccmp 23303 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 |
This theorem is referenced by: xkoopn 23506 xkouni 23516 xkoccn 23536 xkoco1cn 23574 xkoco2cn 23575 xkococn 23577 xkoinjcn 23604 |
Copyright terms: Public domain | W3C validator |