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Mirrors > Home > MPE Home > Th. List > hof2val | Structured version Visualization version GIF version |
Description: The morphism part of the Hom functor, for morphisms β¨π, πβ©:β¨π, πβ©βΆβ¨π, πβ© (which since the first argument is contravariant means morphisms π:πβΆπ and π:πβΆπ), yields a function (a morphism of SetCat) mapping β:πβΆπ to π β β β π:πβΆπ. (Contributed by Mario Carneiro, 15-Jan-2017.) |
Ref | Expression |
---|---|
hofval.m | β’ π = (HomFβπΆ) |
hofval.c | β’ (π β πΆ β Cat) |
hof1.b | β’ π΅ = (BaseβπΆ) |
hof1.h | β’ π» = (Hom βπΆ) |
hof1.x | β’ (π β π β π΅) |
hof1.y | β’ (π β π β π΅) |
hof2.z | β’ (π β π β π΅) |
hof2.w | β’ (π β π β π΅) |
hof2.o | β’ Β· = (compβπΆ) |
hof2.f | β’ (π β πΉ β (ππ»π)) |
hof2.g | β’ (π β πΊ β (ππ»π)) |
Ref | Expression |
---|---|
hof2val | β’ (π β (πΉ(β¨π, πβ©(2nd βπ)β¨π, πβ©)πΊ) = (β β (ππ»π) β¦ ((πΊ(β¨π, πβ© Β· π)β)(β¨π, πβ© Β· π)πΉ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hofval.m | . . 3 β’ π = (HomFβπΆ) | |
2 | hofval.c | . . 3 β’ (π β πΆ β Cat) | |
3 | hof1.b | . . 3 β’ π΅ = (BaseβπΆ) | |
4 | hof1.h | . . 3 β’ π» = (Hom βπΆ) | |
5 | hof1.x | . . 3 β’ (π β π β π΅) | |
6 | hof1.y | . . 3 β’ (π β π β π΅) | |
7 | hof2.z | . . 3 β’ (π β π β π΅) | |
8 | hof2.w | . . 3 β’ (π β π β π΅) | |
9 | hof2.o | . . 3 β’ Β· = (compβπΆ) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | hof2fval 18218 | . 2 β’ (π β (β¨π, πβ©(2nd βπ)β¨π, πβ©) = (π β (ππ»π), π β (ππ»π) β¦ (β β (ππ»π) β¦ ((π(β¨π, πβ© Β· π)β)(β¨π, πβ© Β· π)π)))) |
11 | simplrr 775 | . . . . 5 β’ (((π β§ (π = πΉ β§ π = πΊ)) β§ β β (ππ»π)) β π = πΊ) | |
12 | 11 | oveq1d 7419 | . . . 4 β’ (((π β§ (π = πΉ β§ π = πΊ)) β§ β β (ππ»π)) β (π(β¨π, πβ© Β· π)β) = (πΊ(β¨π, πβ© Β· π)β)) |
13 | simplrl 774 | . . . 4 β’ (((π β§ (π = πΉ β§ π = πΊ)) β§ β β (ππ»π)) β π = πΉ) | |
14 | 12, 13 | oveq12d 7422 | . . 3 β’ (((π β§ (π = πΉ β§ π = πΊ)) β§ β β (ππ»π)) β ((π(β¨π, πβ© Β· π)β)(β¨π, πβ© Β· π)π) = ((πΊ(β¨π, πβ© Β· π)β)(β¨π, πβ© Β· π)πΉ)) |
15 | 14 | mpteq2dva 5241 | . 2 β’ ((π β§ (π = πΉ β§ π = πΊ)) β (β β (ππ»π) β¦ ((π(β¨π, πβ© Β· π)β)(β¨π, πβ© Β· π)π)) = (β β (ππ»π) β¦ ((πΊ(β¨π, πβ© Β· π)β)(β¨π, πβ© Β· π)πΉ))) |
16 | hof2.f | . 2 β’ (π β πΉ β (ππ»π)) | |
17 | hof2.g | . 2 β’ (π β πΊ β (ππ»π)) | |
18 | ovex 7437 | . . . 4 β’ (ππ»π) β V | |
19 | 18 | mptex 7219 | . . 3 β’ (β β (ππ»π) β¦ ((πΊ(β¨π, πβ© Β· π)β)(β¨π, πβ© Β· π)πΉ)) β V |
20 | 19 | a1i 11 | . 2 β’ (π β (β β (ππ»π) β¦ ((πΊ(β¨π, πβ© Β· π)β)(β¨π, πβ© Β· π)πΉ)) β V) |
21 | 10, 15, 16, 17, 20 | ovmpod 7555 | 1 β’ (π β (πΉ(β¨π, πβ©(2nd βπ)β¨π, πβ©)πΊ) = (β β (ππ»π) β¦ ((πΊ(β¨π, πβ© Β· π)β)(β¨π, πβ© Β· π)πΉ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 Vcvv 3468 β¨cop 4629 β¦ cmpt 5224 βcfv 6536 (class class class)co 7404 2nd c2nd 7970 Basecbs 17151 Hom chom 17215 compcco 17216 Catccat 17615 HomFchof 18211 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-hof 18213 |
This theorem is referenced by: hof2 18220 hofcllem 18221 hofcl 18222 yonedalem3b 18242 |
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