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Theorem hof2val 17789
Description: The morphism part of the Hom functor, for morphisms 𝑓, 𝑔⟩:⟨𝑋, 𝑌⟩⟶⟨𝑍, 𝑊 (which since the first argument is contravariant means morphisms 𝑓:𝑍𝑋 and 𝑔:𝑌𝑊), yields a function (a morphism of SetCat) mapping :𝑋𝑌 to 𝑔𝑓:𝑍𝑊. (Contributed by Mario Carneiro, 15-Jan-2017.)
Hypotheses
Ref Expression
hofval.m 𝑀 = (HomF𝐶)
hofval.c (𝜑𝐶 ∈ Cat)
hof1.b 𝐵 = (Base‘𝐶)
hof1.h 𝐻 = (Hom ‘𝐶)
hof1.x (𝜑𝑋𝐵)
hof1.y (𝜑𝑌𝐵)
hof2.z (𝜑𝑍𝐵)
hof2.w (𝜑𝑊𝐵)
hof2.o · = (comp‘𝐶)
hof2.f (𝜑𝐹 ∈ (𝑍𝐻𝑋))
hof2.g (𝜑𝐺 ∈ (𝑌𝐻𝑊))
Assertion
Ref Expression
hof2val (𝜑 → (𝐹(⟨𝑋, 𝑌⟩(2nd𝑀)⟨𝑍, 𝑊⟩)𝐺) = ( ∈ (𝑋𝐻𝑌) ↦ ((𝐺(⟨𝑋, 𝑌· 𝑊))(⟨𝑍, 𝑋· 𝑊)𝐹)))
Distinct variable groups:   𝐵,   ,𝐹   ,𝐺   𝜑,   𝐶,   ,𝐻   ,𝑊   · ,   ,𝑋   ,𝑌   ,𝑍
Allowed substitution hint:   𝑀()

Proof of Theorem hof2val
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hofval.m . . 3 𝑀 = (HomF𝐶)
2 hofval.c . . 3 (𝜑𝐶 ∈ Cat)
3 hof1.b . . 3 𝐵 = (Base‘𝐶)
4 hof1.h . . 3 𝐻 = (Hom ‘𝐶)
5 hof1.x . . 3 (𝜑𝑋𝐵)
6 hof1.y . . 3 (𝜑𝑌𝐵)
7 hof2.z . . 3 (𝜑𝑍𝐵)
8 hof2.w . . 3 (𝜑𝑊𝐵)
9 hof2.o . . 3 · = (comp‘𝐶)
101, 2, 3, 4, 5, 6, 7, 8, 9hof2fval 17788 . 2 (𝜑 → (⟨𝑋, 𝑌⟩(2nd𝑀)⟨𝑍, 𝑊⟩) = (𝑓 ∈ (𝑍𝐻𝑋), 𝑔 ∈ (𝑌𝐻𝑊) ↦ ( ∈ (𝑋𝐻𝑌) ↦ ((𝑔(⟨𝑋, 𝑌· 𝑊))(⟨𝑍, 𝑋· 𝑊)𝑓))))
11 simplrr 778 . . . . 5 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ ∈ (𝑋𝐻𝑌)) → 𝑔 = 𝐺)
1211oveq1d 7247 . . . 4 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ ∈ (𝑋𝐻𝑌)) → (𝑔(⟨𝑋, 𝑌· 𝑊)) = (𝐺(⟨𝑋, 𝑌· 𝑊)))
13 simplrl 777 . . . 4 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ ∈ (𝑋𝐻𝑌)) → 𝑓 = 𝐹)
1412, 13oveq12d 7250 . . 3 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ ∈ (𝑋𝐻𝑌)) → ((𝑔(⟨𝑋, 𝑌· 𝑊))(⟨𝑍, 𝑋· 𝑊)𝑓) = ((𝐺(⟨𝑋, 𝑌· 𝑊))(⟨𝑍, 𝑋· 𝑊)𝐹))
1514mpteq2dva 5165 . 2 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → ( ∈ (𝑋𝐻𝑌) ↦ ((𝑔(⟨𝑋, 𝑌· 𝑊))(⟨𝑍, 𝑋· 𝑊)𝑓)) = ( ∈ (𝑋𝐻𝑌) ↦ ((𝐺(⟨𝑋, 𝑌· 𝑊))(⟨𝑍, 𝑋· 𝑊)𝐹)))
16 hof2.f . 2 (𝜑𝐹 ∈ (𝑍𝐻𝑋))
17 hof2.g . 2 (𝜑𝐺 ∈ (𝑌𝐻𝑊))
18 ovex 7265 . . . 4 (𝑋𝐻𝑌) ∈ V
1918mptex 7058 . . 3 ( ∈ (𝑋𝐻𝑌) ↦ ((𝐺(⟨𝑋, 𝑌· 𝑊))(⟨𝑍, 𝑋· 𝑊)𝐹)) ∈ V
2019a1i 11 . 2 (𝜑 → ( ∈ (𝑋𝐻𝑌) ↦ ((𝐺(⟨𝑋, 𝑌· 𝑊))(⟨𝑍, 𝑋· 𝑊)𝐹)) ∈ V)
2110, 15, 16, 17, 20ovmpod 7380 1 (𝜑 → (𝐹(⟨𝑋, 𝑌⟩(2nd𝑀)⟨𝑍, 𝑊⟩)𝐺) = ( ∈ (𝑋𝐻𝑌) ↦ ((𝐺(⟨𝑋, 𝑌· 𝑊))(⟨𝑍, 𝑋· 𝑊)𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2111  Vcvv 3421  cop 4562  cmpt 5150  cfv 6398  (class class class)co 7232  2nd c2nd 7779  Basecbs 16785  Hom chom 16838  compcco 16839  Catccat 17192  HomFchof 17781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-rep 5194  ax-sep 5207  ax-nul 5214  ax-pow 5273  ax-pr 5337  ax-un 7542
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-reu 3069  df-rab 3071  df-v 3423  df-sbc 3710  df-csb 3827  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4253  df-if 4455  df-pw 4530  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4835  df-iun 4921  df-br 5069  df-opab 5131  df-mpt 5151  df-id 5470  df-xp 5572  df-rel 5573  df-cnv 5574  df-co 5575  df-dm 5576  df-rn 5577  df-res 5578  df-ima 5579  df-iota 6356  df-fun 6400  df-fn 6401  df-f 6402  df-f1 6403  df-fo 6404  df-f1o 6405  df-fv 6406  df-ov 7235  df-oprab 7236  df-mpo 7237  df-1st 7780  df-2nd 7781  df-hof 17783
This theorem is referenced by:  hof2  17790  hofcllem  17791  hofcl  17792  yonedalem3b  17812
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