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

Theorem fnfuc 17837
Description: The FuncCat operation is a well-defined function on categories. (Contributed by Mario Carneiro, 12-Jan-2017.)
Assertion
Ref Expression
fnfuc FuncCat Fn (Cat Γ— Cat)

Proof of Theorem fnfuc
Dummy variables π‘Ž 𝑏 𝑓 𝑔 β„Ž 𝑑 𝑒 𝑣 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-fuc 17836 . 2 FuncCat = (𝑑 ∈ Cat, 𝑒 ∈ Cat ↦ {⟨(Baseβ€˜ndx), (𝑑 Func 𝑒)⟩, ⟨(Hom β€˜ndx), (𝑑 Nat 𝑒)⟩, ⟨(compβ€˜ndx), (𝑣 ∈ ((𝑑 Func 𝑒) Γ— (𝑑 Func 𝑒)), β„Ž ∈ (𝑑 Func 𝑒) ↦ ⦋(1st β€˜π‘£) / π‘“β¦Œβ¦‹(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (𝑔(𝑑 Nat 𝑒)β„Ž), π‘Ž ∈ (𝑓(𝑑 Nat 𝑒)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‘) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘’)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))))⟩})
2 tpex 7682 . 2 {⟨(Baseβ€˜ndx), (𝑑 Func 𝑒)⟩, ⟨(Hom β€˜ndx), (𝑑 Nat 𝑒)⟩, ⟨(compβ€˜ndx), (𝑣 ∈ ((𝑑 Func 𝑒) Γ— (𝑑 Func 𝑒)), β„Ž ∈ (𝑑 Func 𝑒) ↦ ⦋(1st β€˜π‘£) / π‘“β¦Œβ¦‹(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (𝑔(𝑑 Nat 𝑒)β„Ž), π‘Ž ∈ (𝑓(𝑑 Nat 𝑒)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‘) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘’)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))))⟩} ∈ V
31, 2fnmpoi 8003 1 FuncCat Fn (Cat Γ— Cat)
Colors of variables: wff setvar class
Syntax hints:  β¦‹csb 3856  {ctp 4591  βŸ¨cop 4593   ↦ cmpt 5189   Γ— cxp 5632   Fn wfn 6492  β€˜cfv 6497  (class class class)co 7358   ∈ cmpo 7360  1st c1st 7920  2nd c2nd 7921  ndxcnx 17070  Basecbs 17088  Hom chom 17149  compcco 17150  Catccat 17549   Func cfunc 17745   Nat cnat 17833   FuncCat cfuc 17834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-fuc 17836
This theorem is referenced by:  fucbas  17853  fuchom  17854  fuchomOLD  17855
  Copyright terms: Public domain W3C validator