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Theorem hof2 17105
 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:𝑍⟶𝑋 and 𝑔:𝑌⟶𝑊), yields a function (a morphism of SetCat) mapping ℎ:𝑋⟶𝑌 to 𝑔 ∘ ℎ ∘ 𝑓:𝑍⟶𝑊. (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))
hof2.k (𝜑𝐾 ∈ (𝑋𝐻𝑌))
Assertion
Ref Expression
hof2 (𝜑 → ((𝐹(⟨𝑋, 𝑌⟩(2nd𝑀)⟨𝑍, 𝑊⟩)𝐺)‘𝐾) = ((𝐺(⟨𝑋, 𝑌· 𝑊)𝐾)(⟨𝑍, 𝑋· 𝑊)𝐹))

Proof of Theorem hof2
Dummy variable is 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)
10 hof2.f . . 3 (𝜑𝐹 ∈ (𝑍𝐻𝑋))
11 hof2.g . . 3 (𝜑𝐺 ∈ (𝑌𝐻𝑊))
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11hof2val 17104 . 2 (𝜑 → (𝐹(⟨𝑋, 𝑌⟩(2nd𝑀)⟨𝑍, 𝑊⟩)𝐺) = ( ∈ (𝑋𝐻𝑌) ↦ ((𝐺(⟨𝑋, 𝑌· 𝑊))(⟨𝑍, 𝑋· 𝑊)𝐹)))
13 simpr 471 . . . 4 ((𝜑 = 𝐾) → = 𝐾)
1413oveq2d 6809 . . 3 ((𝜑 = 𝐾) → (𝐺(⟨𝑋, 𝑌· 𝑊)) = (𝐺(⟨𝑋, 𝑌· 𝑊)𝐾))
1514oveq1d 6808 . 2 ((𝜑 = 𝐾) → ((𝐺(⟨𝑋, 𝑌· 𝑊))(⟨𝑍, 𝑋· 𝑊)𝐹) = ((𝐺(⟨𝑋, 𝑌· 𝑊)𝐾)(⟨𝑍, 𝑋· 𝑊)𝐹))
16 hof2.k . 2 (𝜑𝐾 ∈ (𝑋𝐻𝑌))
17 ovexd 6825 . 2 (𝜑 → ((𝐺(⟨𝑋, 𝑌· 𝑊)𝐾)(⟨𝑍, 𝑋· 𝑊)𝐹) ∈ V)
1812, 15, 16, 17fvmptd 6430 1 (𝜑 → ((𝐹(⟨𝑋, 𝑌⟩(2nd𝑀)⟨𝑍, 𝑊⟩)𝐺)‘𝐾) = ((𝐺(⟨𝑋, 𝑌· 𝑊)𝐾)(⟨𝑍, 𝑋· 𝑊)𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145  Vcvv 3351  ⟨cop 4322  ‘cfv 6031  (class class class)co 6793  2nd c2nd 7314  Basecbs 16064  Hom chom 16160  compcco 16161  Catccat 16532  HomFchof 17096 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-hof 17098 This theorem is referenced by:  yon12  17113  yon2  17114
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