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Theorem fnfuc 17900
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 17899 . 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 7728 . 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 8050 1 FuncCat Fn (Cat Γ— Cat)
Colors of variables: wff setvar class
Syntax hints:  β¦‹csb 3886  {ctp 4625  βŸ¨cop 4627   ↦ cmpt 5222   Γ— cxp 5665   Fn wfn 6529  β€˜cfv 6534  (class class class)co 7402   ∈ cmpo 7404  1st c1st 7967  2nd c2nd 7968  ndxcnx 17127  Basecbs 17145  Hom chom 17209  compcco 17210  Catccat 17609   Func cfunc 17805   Nat cnat 17896   FuncCat cfuc 17897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-tp 4626  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-oprab 7406  df-mpo 7407  df-1st 7969  df-2nd 7970  df-fuc 17899
This theorem is referenced by:  fucbas  17916  fuchom  17917  fuchomOLD  17918
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