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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 18145 | . . 3 β’ Yon = (π β Cat β¦ (β¨π, (oppCatβπ)β© curryF (HomFβ(oppCatβπ)))) | |
3 | simpr 486 | . . . . 5 β’ ((π β§ π = πΆ) β π = πΆ) | |
4 | 3 | fveq2d 6847 | . . . . . 6 β’ ((π β§ π = πΆ) β (oppCatβπ) = (oppCatβπΆ)) |
5 | yonval.o | . . . . . 6 β’ π = (oppCatβπΆ) | |
6 | 4, 5 | eqtr4di 2791 | . . . . 5 β’ ((π β§ π = πΆ) β (oppCatβπ) = π) |
7 | 3, 6 | opeq12d 4839 | . . . 4 β’ ((π β§ π = πΆ) β β¨π, (oppCatβπ)β© = β¨πΆ, πβ©) |
8 | 6 | fveq2d 6847 | . . . . 5 β’ ((π β§ π = πΆ) β (HomFβ(oppCatβπ)) = (HomFβπ)) |
9 | yonval.m | . . . . 5 β’ π = (HomFβπ) | |
10 | 8, 9 | eqtr4di 2791 | . . . 4 β’ ((π β§ π = πΆ) β (HomFβ(oppCatβπ)) = π) |
11 | 7, 10 | oveq12d 7376 | . . 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 7393 | . . 3 β’ (π β (β¨πΆ, πβ© curryF π) β V) | |
14 | 2, 11, 12, 13 | fvmptd2 6957 | . 2 β’ (π β (YonβπΆ) = (β¨πΆ, πβ© curryF π)) |
15 | 1, 14 | eqtrid 2785 | 1 β’ (π β π = (β¨πΆ, πβ© curryF π)) |
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 Catccat 17549 oppCatcoppc 17596 curryF ccurf 18104 HomFchof 18142 Yoncyon 18143 |
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 |
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-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-op 4594 df-uni 4867 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-iota 6449 df-fun 6499 df-fv 6505 df-ov 7361 df-yon 18145 |
This theorem is referenced by: yoncl 18156 yon11 18158 yon12 18159 yon2 18160 yonpropd 18162 oppcyon 18163 |
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