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Theorem hof2val 16719
 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))
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 16718 . 2 (𝜑 → (⟨𝑋, 𝑌⟩(2nd𝑀)⟨𝑍, 𝑊⟩) = (𝑓 ∈ (𝑍𝐻𝑋), 𝑔 ∈ (𝑌𝐻𝑊) ↦ ( ∈ (𝑋𝐻𝑌) ↦ ((𝑔(⟨𝑋, 𝑌· 𝑊))(⟨𝑍, 𝑋· 𝑊)𝑓))))
11 simplrr 797 . . . . 5 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ ∈ (𝑋𝐻𝑌)) → 𝑔 = 𝐺)
1211oveq1d 6564 . . . 4 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ ∈ (𝑋𝐻𝑌)) → (𝑔(⟨𝑋, 𝑌· 𝑊)) = (𝐺(⟨𝑋, 𝑌· 𝑊)))
13 simplrl 796 . . . 4 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ ∈ (𝑋𝐻𝑌)) → 𝑓 = 𝐹)
1412, 13oveq12d 6567 . . 3 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ ∈ (𝑋𝐻𝑌)) → ((𝑔(⟨𝑋, 𝑌· 𝑊))(⟨𝑍, 𝑋· 𝑊)𝑓) = ((𝐺(⟨𝑋, 𝑌· 𝑊))(⟨𝑍, 𝑋· 𝑊)𝐹))
1514mpteq2dva 4672 . 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 6577 . . . 4 (𝑋𝐻𝑌) ∈ V
1918mptex 6390 . . 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, 20ovmpt2d 6686 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 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  2nd c2nd 7058  Basecbs 15695  Hom chom 15779  compcco 15780  Catccat 16148  HomFchof 16711 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-hof 16713 This theorem is referenced by:  hof2  16720  hofcllem  16721  hofcl  16722  yonedalem3b  16742
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