Theorem evlf2val 17918

Description: Value of the evaluation natural transformation at an object. (Contributed by Mario Carneiro, 12-Jan-2017.)

Hypotheses

Ref	Expression
evlfval.e	⊢ 𝐸 = (𝐶 evalF 𝐷)
evlfval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
evlfval.d	⊢ (𝜑 → 𝐷 ∈ Cat)
evlfval.b	⊢ 𝐵 = (Base‘𝐶)
evlfval.h	⊢ 𝐻 = (Hom ‘𝐶)
evlfval.o	⊢ · = (comp‘𝐷)
evlfval.n	⊢ 𝑁 = (𝐶 Nat 𝐷)
evlf2.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
evlf2.g	⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
evlf2.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
evlf2.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
evlf2.l	⊢ 𝐿 = (⟨𝐹, 𝑋⟩(2nd ‘𝐸)⟨𝐺, 𝑌⟩)
evlf2val.a	⊢ (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺))
evlf2val.k	⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌))

Assertion

Ref	Expression
evlf2val	⊢ (𝜑 → (𝐴𝐿𝐾) = ((𝐴‘𝑌)(⟨((1st ‘𝐹)‘𝑋), ((1st ‘𝐹)‘𝑌)⟩ · ((1st ‘𝐺)‘𝑌))((𝑋(2nd ‘𝐹)𝑌)‘𝐾)))

Proof of Theorem evlf2val

Dummy variables 𝑎 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		evlfval.e	. . 3 ⊢ 𝐸 = (𝐶 evalF 𝐷)
2		evlfval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
3		evlfval.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
4		evlfval.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
5		evlfval.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
6		evlfval.o	. . 3 ⊢ · = (comp‘𝐷)
7		evlfval.n	. . 3 ⊢ 𝑁 = (𝐶 Nat 𝐷)
8		evlf2.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
9		evlf2.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
10		evlf2.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
11		evlf2.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
12		evlf2.l	. . 3 ⊢ 𝐿 = (⟨𝐹, 𝑋⟩(2nd ‘𝐸)⟨𝐺, 𝑌⟩)
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	evlf2 17917	. 2 ⊢ (𝜑 → 𝐿 = (𝑎 ∈ (𝐹𝑁𝐺), 𝑔 ∈ (𝑋𝐻𝑌) ↦ ((𝑎‘𝑌)(⟨((1st ‘𝐹)‘𝑋), ((1st ‘𝐹)‘𝑌)⟩ · ((1st ‘𝐺)‘𝑌))((𝑋(2nd ‘𝐹)𝑌)‘𝑔))))
14		simprl 767	. . . 4 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑔 = 𝐾)) → 𝑎 = 𝐴)
15	14	fveq1d 6770	. . 3 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑔 = 𝐾)) → (𝑎‘𝑌) = (𝐴‘𝑌))
16		simprr 769	. . . 4 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑔 = 𝐾)) → 𝑔 = 𝐾)
17	16	fveq2d 6772	. . 3 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑔 = 𝐾)) → ((𝑋(2nd ‘𝐹)𝑌)‘𝑔) = ((𝑋(2nd ‘𝐹)𝑌)‘𝐾))
18	15, 17	oveq12d 7286	. 2 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑔 = 𝐾)) → ((𝑎‘𝑌)(⟨((1st ‘𝐹)‘𝑋), ((1st ‘𝐹)‘𝑌)⟩ · ((1st ‘𝐺)‘𝑌))((𝑋(2nd ‘𝐹)𝑌)‘𝑔)) = ((𝐴‘𝑌)(⟨((1st ‘𝐹)‘𝑋), ((1st ‘𝐹)‘𝑌)⟩ · ((1st ‘𝐺)‘𝑌))((𝑋(2nd ‘𝐹)𝑌)‘𝐾)))
19		evlf2val.a	. 2 ⊢ (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺))
20		evlf2val.k	. 2 ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌))
21		ovexd 7303	. 2 ⊢ (𝜑 → ((𝐴‘𝑌)(⟨((1st ‘𝐹)‘𝑋), ((1st ‘𝐹)‘𝑌)⟩ · ((1st ‘𝐺)‘𝑌))((𝑋(2nd ‘𝐹)𝑌)‘𝐾)) ∈ V)
22	13, 18, 19, 20, 21	ovmpod 7416	1 ⊢ (𝜑 → (𝐴𝐿𝐾) = ((𝐴‘𝑌)(⟨((1st ‘𝐹)‘𝑋), ((1st ‘𝐹)‘𝑌)⟩ · ((1st ‘𝐺)‘𝑌))((𝑋(2nd ‘𝐹)𝑌)‘𝐾)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2109  Vcvv 3430  ⟨cop 4572  ‘cfv 6430  (class class class)co 7268  1^st c1st 7815  2^nd c2nd 7816  Basecbs 16893  Hom chom 16954  compcco 16955  Catccat 17354   Func cfunc 17550   Nat cnat 17638   eval_F cevlf 17908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-ov 7271  df-oprab 7272  df-mpo 7273  df-1st 7817  df-2nd 7818  df-evlf 17912

This theorem is referenced by:  evlfcllem  17920  evlfcl  17921  uncf2  17936  yonedalem3b  17978