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Mirrors > Home > MPE Home > Th. List > yonval | Structured version Visualization version GIF version |
Description: Value of the Yoneda embedding. (Contributed by Mario Carneiro, 17-Jan-2017.) |
Ref | Expression |
---|---|
yonval.y | β’ π = (YonβπΆ) |
yonval.c | β’ (π β πΆ β Cat) |
yonval.o | β’ π = (oppCatβπΆ) |
yonval.m | β’ π = (HomFβπ) |
Ref | Expression |
---|---|
yonval | β’ (π β π = (β¨πΆ, πβ© curryF π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | yonval.y | . 2 β’ π = (YonβπΆ) | |
2 | df-yon 18243 | . . 3 β’ Yon = (π β Cat β¦ (β¨π, (oppCatβπ)β© curryF (HomFβ(oppCatβπ)))) | |
3 | simpr 484 | . . . . 5 β’ ((π β§ π = πΆ) β π = πΆ) | |
4 | 3 | fveq2d 6901 | . . . . . 6 β’ ((π β§ π = πΆ) β (oppCatβπ) = (oppCatβπΆ)) |
5 | yonval.o | . . . . . 6 β’ π = (oppCatβπΆ) | |
6 | 4, 5 | eqtr4di 2786 | . . . . 5 β’ ((π β§ π = πΆ) β (oppCatβπ) = π) |
7 | 3, 6 | opeq12d 4882 | . . . 4 β’ ((π β§ π = πΆ) β β¨π, (oppCatβπ)β© = β¨πΆ, πβ©) |
8 | 6 | fveq2d 6901 | . . . . 5 β’ ((π β§ π = πΆ) β (HomFβ(oppCatβπ)) = (HomFβπ)) |
9 | yonval.m | . . . . 5 β’ π = (HomFβπ) | |
10 | 8, 9 | eqtr4di 2786 | . . . 4 β’ ((π β§ π = πΆ) β (HomFβ(oppCatβπ)) = π) |
11 | 7, 10 | oveq12d 7438 | . . 3 β’ ((π β§ π = πΆ) β (β¨π, (oppCatβπ)β© curryF (HomFβ(oppCatβπ))) = (β¨πΆ, πβ© curryF π)) |
12 | yonval.c | . . 3 β’ (π β πΆ β Cat) | |
13 | ovexd 7455 | . . 3 β’ (π β (β¨πΆ, πβ© curryF π) β V) | |
14 | 2, 11, 12, 13 | fvmptd2 7013 | . 2 β’ (π β (YonβπΆ) = (β¨πΆ, πβ© curryF π)) |
15 | 1, 14 | eqtrid 2780 | 1 β’ (π β π = (β¨πΆ, πβ© curryF π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 Vcvv 3471 β¨cop 4635 βcfv 6548 (class class class)co 7420 Catccat 17644 oppCatcoppc 17691 curryF ccurf 18202 HomFchof 18240 Yoncyon 18241 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6500 df-fun 6550 df-fv 6556 df-ov 7423 df-yon 18243 |
This theorem is referenced by: yoncl 18254 yon11 18256 yon12 18257 yon2 18258 yonpropd 18260 oppcyon 18261 |
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