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Theorem evlf2val 18177
Description: Value of the evaluation natural transformation at an object. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
evlfval.e ๐ธ = (๐ถ evalF ๐ท)
evlfval.c (๐œ‘ โ†’ ๐ถ โˆˆ Cat)
evlfval.d (๐œ‘ โ†’ ๐ท โˆˆ Cat)
evlfval.b ๐ต = (Baseโ€˜๐ถ)
evlfval.h ๐ป = (Hom โ€˜๐ถ)
evlfval.o ยท = (compโ€˜๐ท)
evlfval.n ๐‘ = (๐ถ Nat ๐ท)
evlf2.f (๐œ‘ โ†’ ๐น โˆˆ (๐ถ Func ๐ท))
evlf2.g (๐œ‘ โ†’ ๐บ โˆˆ (๐ถ Func ๐ท))
evlf2.x (๐œ‘ โ†’ ๐‘‹ โˆˆ ๐ต)
evlf2.y (๐œ‘ โ†’ ๐‘Œ โˆˆ ๐ต)
evlf2.l ๐ฟ = (โŸจ๐น, ๐‘‹โŸฉ(2nd โ€˜๐ธ)โŸจ๐บ, ๐‘ŒโŸฉ)
evlf2val.a (๐œ‘ โ†’ ๐ด โˆˆ (๐น๐‘๐บ))
evlf2val.k (๐œ‘ โ†’ ๐พ โˆˆ (๐‘‹๐ป๐‘Œ))
Assertion
Ref Expression
evlf2val (๐œ‘ โ†’ (๐ด๐ฟ๐พ) = ((๐ดโ€˜๐‘Œ)(โŸจ((1st โ€˜๐น)โ€˜๐‘‹), ((1st โ€˜๐น)โ€˜๐‘Œ)โŸฉ ยท ((1st โ€˜๐บ)โ€˜๐‘Œ))((๐‘‹(2nd โ€˜๐น)๐‘Œ)โ€˜๐พ)))

Proof of Theorem evlf2val
Dummy variables ๐‘Ž ๐‘” are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 evlfval.e . . 3 ๐ธ = (๐ถ evalF ๐ท)
2 evlfval.c . . 3 (๐œ‘ โ†’ ๐ถ โˆˆ Cat)
3 evlfval.d . . 3 (๐œ‘ โ†’ ๐ท โˆˆ Cat)
4 evlfval.b . . 3 ๐ต = (Baseโ€˜๐ถ)
5 evlfval.h . . 3 ๐ป = (Hom โ€˜๐ถ)
6 evlfval.o . . 3 ยท = (compโ€˜๐ท)
7 evlfval.n . . 3 ๐‘ = (๐ถ Nat ๐ท)
8 evlf2.f . . 3 (๐œ‘ โ†’ ๐น โˆˆ (๐ถ Func ๐ท))
9 evlf2.g . . 3 (๐œ‘ โ†’ ๐บ โˆˆ (๐ถ Func ๐ท))
10 evlf2.x . . 3 (๐œ‘ โ†’ ๐‘‹ โˆˆ ๐ต)
11 evlf2.y . . 3 (๐œ‘ โ†’ ๐‘Œ โˆˆ ๐ต)
12 evlf2.l . . 3 ๐ฟ = (โŸจ๐น, ๐‘‹โŸฉ(2nd โ€˜๐ธ)โŸจ๐บ, ๐‘ŒโŸฉ)
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12evlf2 18176 . 2 (๐œ‘ โ†’ ๐ฟ = (๐‘Ž โˆˆ (๐น๐‘๐บ), ๐‘” โˆˆ (๐‘‹๐ป๐‘Œ) โ†ฆ ((๐‘Žโ€˜๐‘Œ)(โŸจ((1st โ€˜๐น)โ€˜๐‘‹), ((1st โ€˜๐น)โ€˜๐‘Œ)โŸฉ ยท ((1st โ€˜๐บ)โ€˜๐‘Œ))((๐‘‹(2nd โ€˜๐น)๐‘Œ)โ€˜๐‘”))))
14 simprl 768 . . . 4 ((๐œ‘ โˆง (๐‘Ž = ๐ด โˆง ๐‘” = ๐พ)) โ†’ ๐‘Ž = ๐ด)
1514fveq1d 6893 . . 3 ((๐œ‘ โˆง (๐‘Ž = ๐ด โˆง ๐‘” = ๐พ)) โ†’ (๐‘Žโ€˜๐‘Œ) = (๐ดโ€˜๐‘Œ))
16 simprr 770 . . . 4 ((๐œ‘ โˆง (๐‘Ž = ๐ด โˆง ๐‘” = ๐พ)) โ†’ ๐‘” = ๐พ)
1716fveq2d 6895 . . 3 ((๐œ‘ โˆง (๐‘Ž = ๐ด โˆง ๐‘” = ๐พ)) โ†’ ((๐‘‹(2nd โ€˜๐น)๐‘Œ)โ€˜๐‘”) = ((๐‘‹(2nd โ€˜๐น)๐‘Œ)โ€˜๐พ))
1815, 17oveq12d 7430 . 2 ((๐œ‘ โˆง (๐‘Ž = ๐ด โˆง ๐‘” = ๐พ)) โ†’ ((๐‘Žโ€˜๐‘Œ)(โŸจ((1st โ€˜๐น)โ€˜๐‘‹), ((1st โ€˜๐น)โ€˜๐‘Œ)โŸฉ ยท ((1st โ€˜๐บ)โ€˜๐‘Œ))((๐‘‹(2nd โ€˜๐น)๐‘Œ)โ€˜๐‘”)) = ((๐ดโ€˜๐‘Œ)(โŸจ((1st โ€˜๐น)โ€˜๐‘‹), ((1st โ€˜๐น)โ€˜๐‘Œ)โŸฉ ยท ((1st โ€˜๐บ)โ€˜๐‘Œ))((๐‘‹(2nd โ€˜๐น)๐‘Œ)โ€˜๐พ)))
19 evlf2val.a . 2 (๐œ‘ โ†’ ๐ด โˆˆ (๐น๐‘๐บ))
20 evlf2val.k . 2 (๐œ‘ โ†’ ๐พ โˆˆ (๐‘‹๐ป๐‘Œ))
21 ovexd 7447 . 2 (๐œ‘ โ†’ ((๐ดโ€˜๐‘Œ)(โŸจ((1st โ€˜๐น)โ€˜๐‘‹), ((1st โ€˜๐น)โ€˜๐‘Œ)โŸฉ ยท ((1st โ€˜๐บ)โ€˜๐‘Œ))((๐‘‹(2nd โ€˜๐น)๐‘Œ)โ€˜๐พ)) โˆˆ V)
2213, 18, 19, 20, 21ovmpod 7563 1 (๐œ‘ โ†’ (๐ด๐ฟ๐พ) = ((๐ดโ€˜๐‘Œ)(โŸจ((1st โ€˜๐น)โ€˜๐‘‹), ((1st โ€˜๐น)โ€˜๐‘Œ)โŸฉ ยท ((1st โ€˜๐บ)โ€˜๐‘Œ))((๐‘‹(2nd โ€˜๐น)๐‘Œ)โ€˜๐พ)))
Colors of variables: wff setvar class
Syntax hints:   โ†’ wi 4   โˆง wa 395   = wceq 1540   โˆˆ wcel 2105  Vcvv 3473  โŸจcop 4634  โ€˜cfv 6543  (class class class)co 7412  1st c1st 7976  2nd c2nd 7977  Basecbs 17149  Hom chom 17213  compcco 17214  Catccat 17613   Func cfunc 17809   Nat cnat 17897   evalF cevlf 18167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  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-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7978  df-2nd 7979  df-evlf 18171
This theorem is referenced by:  evlfcllem  18179  evlfcl  18180  uncf2  18195  yonedalem3b  18237
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