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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 17899 | . 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 7728 | . 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 8050 | 1 β’ FuncCat Fn (Cat Γ Cat) |
Colors of variables: wff setvar class |
Syntax hints: β¦csb 3886 {ctp 4625 β¨cop 4627 β¦ cmpt 5222 Γ cxp 5665 Fn wfn 6529 βcfv 6534 (class class class)co 7402 β cmpo 7404 1st c1st 7967 2nd c2nd 7968 ndxcnx 17127 Basecbs 17145 Hom chom 17209 compcco 17210 Catccat 17609 Func cfunc 17805 Nat cnat 17896 FuncCat cfuc 17897 |
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-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-sn 4622 df-pr 4624 df-tp 4626 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-oprab 7406 df-mpo 7407 df-1st 7969 df-2nd 7970 df-fuc 17899 |
This theorem is referenced by: fucbas 17916 fuchom 17917 fuchomOLD 17918 |
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