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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 17836 | . 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 7682 | . 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 8003 | 1 β’ FuncCat Fn (Cat Γ Cat) |
Colors of variables: wff setvar class |
Syntax hints: β¦csb 3856 {ctp 4591 β¨cop 4593 β¦ cmpt 5189 Γ cxp 5632 Fn wfn 6492 βcfv 6497 (class class class)co 7358 β cmpo 7360 1st c1st 7920 2nd c2nd 7921 ndxcnx 17070 Basecbs 17088 Hom chom 17149 compcco 17150 Catccat 17549 Func cfunc 17745 Nat cnat 17833 FuncCat cfuc 17834 |
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 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
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 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-fuc 17836 |
This theorem is referenced by: fucbas 17853 fuchom 17854 fuchomOLD 17855 |
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