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Mirrors > Home > MPE Home > Th. List > fnfuc | Structured version Visualization version GIF version |
Description: The FuncCat operation is a well-defined function on categories. (Contributed by Mario Carneiro, 12-Jan-2017.) |
Ref | Expression |
---|---|
fnfuc | β’ FuncCat Fn (Cat Γ Cat) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fuc 17927 | . 2 β’ FuncCat = (π‘ β Cat, π’ β Cat β¦ {β¨(Baseβndx), (π‘ Func π’)β©, β¨(Hom βndx), (π‘ Nat π’)β©, β¨(compβndx), (π£ β ((π‘ Func π’) Γ (π‘ Func π’)), β β (π‘ Func π’) β¦ β¦(1st βπ£) / πβ¦β¦(2nd βπ£) / πβ¦(π β (π(π‘ Nat π’)β), π β (π(π‘ Nat π’)π) β¦ (π₯ β (Baseβπ‘) β¦ ((πβπ₯)(β¨((1st βπ)βπ₯), ((1st βπ)βπ₯)β©(compβπ’)((1st ββ)βπ₯))(πβπ₯)))))β©}) | |
2 | tpex 7743 | . 2 β’ {β¨(Baseβndx), (π‘ Func π’)β©, β¨(Hom βndx), (π‘ Nat π’)β©, β¨(compβndx), (π£ β ((π‘ Func π’) Γ (π‘ Func π’)), β β (π‘ Func π’) β¦ β¦(1st βπ£) / πβ¦β¦(2nd βπ£) / πβ¦(π β (π(π‘ Nat π’)β), π β (π(π‘ Nat π’)π) β¦ (π₯ β (Baseβπ‘) β¦ ((πβπ₯)(β¨((1st βπ)βπ₯), ((1st βπ)βπ₯)β©(compβπ’)((1st ββ)βπ₯))(πβπ₯)))))β©} β V | |
3 | 1, 2 | fnmpoi 8068 | 1 β’ FuncCat Fn (Cat Γ Cat) |
Colors of variables: wff setvar class |
Syntax hints: β¦csb 3890 {ctp 4628 β¨cop 4630 β¦ cmpt 5225 Γ cxp 5670 Fn wfn 6537 βcfv 6542 (class class class)co 7414 β cmpo 7416 1st c1st 7985 2nd c2nd 7986 ndxcnx 17155 Basecbs 17173 Hom chom 17237 compcco 17238 Catccat 17637 Func cfunc 17833 Nat cnat 17924 FuncCat cfuc 17925 |
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 5293 ax-nul 5300 ax-pr 5423 ax-un 7734 |
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-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 df-fuc 17927 |
This theorem is referenced by: fucbas 17944 fuchom 17945 fuchomOLD 17946 |
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