MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hof2val Structured version   Visualization version   GIF version

Theorem hof2val 17103
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 17102 . 2 (𝜑 → (⟨𝑋, 𝑌⟩(2nd𝑀)⟨𝑍, 𝑊⟩) = (𝑓 ∈ (𝑍𝐻𝑋), 𝑔 ∈ (𝑌𝐻𝑊) ↦ ( ∈ (𝑋𝐻𝑌) ↦ ((𝑔(⟨𝑋, 𝑌· 𝑊))(⟨𝑍, 𝑋· 𝑊)𝑓))))
11 simplrr 755 . . . . 5 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ ∈ (𝑋𝐻𝑌)) → 𝑔 = 𝐺)
1211oveq1d 6807 . . . 4 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ ∈ (𝑋𝐻𝑌)) → (𝑔(⟨𝑋, 𝑌· 𝑊)) = (𝐺(⟨𝑋, 𝑌· 𝑊)))
13 simplrl 754 . . . 4 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ ∈ (𝑋𝐻𝑌)) → 𝑓 = 𝐹)
1412, 13oveq12d 6810 . . 3 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ ∈ (𝑋𝐻𝑌)) → ((𝑔(⟨𝑋, 𝑌· 𝑊))(⟨𝑍, 𝑋· 𝑊)𝑓) = ((𝐺(⟨𝑋, 𝑌· 𝑊))(⟨𝑍, 𝑋· 𝑊)𝐹))
1514mpteq2dva 4876 . 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 6822 . . . 4 (𝑋𝐻𝑌) ∈ V
1918mptex 6629 . . 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 6934 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 382   = wceq 1630  wcel 2144  Vcvv 3349  cop 4320  cmpt 4861  cfv 6031  (class class class)co 6792  2nd c2nd 7313  Basecbs 16063  Hom chom 16159  compcco 16160  Catccat 16531  HomFchof 17095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  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 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-hof 17097
This theorem is referenced by:  hof2  17104  hofcllem  17105  hofcl  17106  yonedalem3b  17126
  Copyright terms: Public domain W3C validator