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Theorem fnfuc 17928
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 17927 . 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 7743 . 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 8068 1 FuncCat Fn (Cat Γ— Cat)
Colors of variables: wff setvar class
Syntax hints:  β¦‹csb 3890  {ctp 4628  βŸ¨cop 4630   ↦ cmpt 5225   Γ— cxp 5670   Fn wfn 6537  β€˜cfv 6542  (class class class)co 7414   ∈ cmpo 7416  1st c1st 7985  2nd c2nd 7986  ndxcnx 17155  Basecbs 17173  Hom chom 17237  compcco 17238  Catccat 17637   Func cfunc 17833   Nat cnat 17924   FuncCat cfuc 17925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-fuc 17927
This theorem is referenced by:  fucbas  17944  fuchom  17945  fuchomOLD  17946
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