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Mirrors > Home > MPE Home > Th. List > setccofval | Structured version Visualization version GIF version |
Description: Composition in the category of sets. (Contributed by Mario Carneiro, 3-Jan-2017.) |
Ref | Expression |
---|---|
setcbas.c | β’ πΆ = (SetCatβπ) |
setcbas.u | β’ (π β π β π) |
setcco.o | β’ Β· = (compβπΆ) |
Ref | Expression |
---|---|
setccofval | β’ (π β Β· = (π£ β (π Γ π), π§ β π β¦ (π β (π§ βm (2nd βπ£)), π β ((2nd βπ£) βm (1st βπ£)) β¦ (π β π)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setcbas.c | . . 3 β’ πΆ = (SetCatβπ) | |
2 | setcbas.u | . . 3 β’ (π β π β π) | |
3 | eqid 2725 | . . . 4 β’ (Hom βπΆ) = (Hom βπΆ) | |
4 | 1, 2, 3 | setchomfval 18067 | . . 3 β’ (π β (Hom βπΆ) = (π₯ β π, π¦ β π β¦ (π¦ βm π₯))) |
5 | eqidd 2726 | . . 3 β’ (π β (π£ β (π Γ π), π§ β π β¦ (π β (π§ βm (2nd βπ£)), π β ((2nd βπ£) βm (1st βπ£)) β¦ (π β π))) = (π£ β (π Γ π), π§ β π β¦ (π β (π§ βm (2nd βπ£)), π β ((2nd βπ£) βm (1st βπ£)) β¦ (π β π)))) | |
6 | 1, 2, 4, 5 | setcval 18065 | . 2 β’ (π β πΆ = {β¨(Baseβndx), πβ©, β¨(Hom βndx), (Hom βπΆ)β©, β¨(compβndx), (π£ β (π Γ π), π§ β π β¦ (π β (π§ βm (2nd βπ£)), π β ((2nd βπ£) βm (1st βπ£)) β¦ (π β π)))β©}) |
7 | catstr 17947 | . 2 β’ {β¨(Baseβndx), πβ©, β¨(Hom βndx), (Hom βπΆ)β©, β¨(compβndx), (π£ β (π Γ π), π§ β π β¦ (π β (π§ βm (2nd βπ£)), π β ((2nd βπ£) βm (1st βπ£)) β¦ (π β π)))β©} Struct β¨1, ;15β© | |
8 | ccoid 17394 | . 2 β’ comp = Slot (compβndx) | |
9 | snsstp3 4817 | . 2 β’ {β¨(compβndx), (π£ β (π Γ π), π§ β π β¦ (π β (π§ βm (2nd βπ£)), π β ((2nd βπ£) βm (1st βπ£)) β¦ (π β π)))β©} β {β¨(Baseβndx), πβ©, β¨(Hom βndx), (Hom βπΆ)β©, β¨(compβndx), (π£ β (π Γ π), π§ β π β¦ (π β (π§ βm (2nd βπ£)), π β ((2nd βπ£) βm (1st βπ£)) β¦ (π β π)))β©} | |
10 | 2, 2 | xpexd 7751 | . . 3 β’ (π β (π Γ π) β V) |
11 | mpoexga 8080 | . . 3 β’ (((π Γ π) β V β§ π β π) β (π£ β (π Γ π), π§ β π β¦ (π β (π§ βm (2nd βπ£)), π β ((2nd βπ£) βm (1st βπ£)) β¦ (π β π))) β V) | |
12 | 10, 2, 11 | syl2anc 582 | . 2 β’ (π β (π£ β (π Γ π), π§ β π β¦ (π β (π§ βm (2nd βπ£)), π β ((2nd βπ£) βm (1st βπ£)) β¦ (π β π))) β V) |
13 | setcco.o | . 2 β’ Β· = (compβπΆ) | |
14 | 6, 7, 8, 9, 12, 13 | strfv3 17173 | 1 β’ (π β Β· = (π£ β (π Γ π), π§ β π β¦ (π β (π§ βm (2nd βπ£)), π β ((2nd βπ£) βm (1st βπ£)) β¦ (π β π)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 Vcvv 3463 {ctp 4628 β¨cop 4630 Γ cxp 5670 β ccom 5676 βcfv 6543 (class class class)co 7416 β cmpo 7418 1st c1st 7989 2nd c2nd 7990 βm cmap 8843 1c1 11139 5c5 12300 ;cdc 12707 ndxcnx 17161 Basecbs 17179 Hom chom 17243 compcco 17244 SetCatcsetc 18063 |
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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 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-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-fz 13517 df-struct 17115 df-slot 17150 df-ndx 17162 df-base 17180 df-hom 17256 df-cco 17257 df-setc 18064 |
This theorem is referenced by: setcco 18071 |
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