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Mirrors > Home > MPE Home > Th. List > hof2 | 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)) |
hof2.k | β’ (π β πΎ β (ππ»π)) |
Ref | Expression |
---|---|
hof2 | β’ (π β ((πΉ(β¨π, πβ©(2nd βπ)β¨π, πβ©)πΊ)βπΎ) = ((πΊ(β¨π, πβ© Β· π)πΎ)(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 | hof2.f | . . 3 β’ (π β πΉ β (ππ»π)) | |
11 | hof2.g | . . 3 β’ (π β πΊ β (ππ»π)) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | hof2val 18150 | . 2 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))) |
13 | simpr 486 | . . . 4 β’ ((π β§ β = πΎ) β β = πΎ) | |
14 | 13 | oveq2d 7374 | . . 3 β’ ((π β§ β = πΎ) β (πΊ(β¨π, πβ© Β· π)β) = (πΊ(β¨π, πβ© Β· π)πΎ)) |
15 | 14 | oveq1d 7373 | . 2 β’ ((π β§ β = πΎ) β ((πΊ(β¨π, πβ© Β· π)β)(β¨π, πβ© Β· π)πΉ) = ((πΊ(β¨π, πβ© Β· π)πΎ)(β¨π, πβ© Β· π)πΉ)) |
16 | hof2.k | . 2 β’ (π β πΎ β (ππ»π)) | |
17 | ovexd 7393 | . 2 β’ (π β ((πΊ(β¨π, πβ© Β· π)πΎ)(β¨π, πβ© Β· π)πΉ) β V) | |
18 | 12, 15, 16, 17 | fvmptd 6956 | 1 β’ (π β ((πΉ(β¨π, πβ©(2nd βπ)β¨π, πβ©)πΊ)βπΎ) = ((πΊ(β¨π, πβ© Β· π)πΎ)(β¨π, πβ© Β· π)πΉ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 Vcvv 3444 β¨cop 4593 βcfv 6497 (class class class)co 7358 2nd c2nd 7921 Basecbs 17088 Hom chom 17149 compcco 17150 Catccat 17549 HomFchof 18142 |
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-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 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-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 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-pw 4563 df-sn 4588 df-pr 4590 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-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-hof 18144 |
This theorem is referenced by: yon12 18159 yon2 18160 |
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