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Theorem fnxpc 18136
Description: The binary product of categories is a two-argument function. (Contributed by Mario Carneiro, 10-Jan-2017.)
Assertion
Ref Expression
fnxpc Γ—c Fn (V Γ— V)

Proof of Theorem fnxpc
Dummy variables 𝑓 𝑏 𝑔 β„Ž π‘Ÿ 𝑠 𝑒 𝑣 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-xpc 18132 . 2 Γ—c = (π‘Ÿ ∈ V, 𝑠 ∈ V ↦ ⦋((Baseβ€˜π‘Ÿ) Γ— (Baseβ€˜π‘ )) / π‘β¦Œβ¦‹(𝑒 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1st β€˜π‘’)(Hom β€˜π‘Ÿ)(1st β€˜π‘£)) Γ— ((2nd β€˜π‘’)(Hom β€˜π‘ )(2nd β€˜π‘£)))) / β„Žβ¦Œ{⟨(Baseβ€˜ndx), π‘βŸ©, ⟨(Hom β€˜ndx), β„ŽβŸ©, ⟨(compβ€˜ndx), (π‘₯ ∈ (𝑏 Γ— 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd β€˜π‘₯)β„Žπ‘¦), 𝑓 ∈ (β„Žβ€˜π‘₯) ↦ ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩(compβ€˜π‘Ÿ)(1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩(compβ€˜π‘ )(2nd β€˜π‘¦))(2nd β€˜π‘“))⟩))⟩})
2 tpex 7728 . . . 4 {⟨(Baseβ€˜ndx), π‘βŸ©, ⟨(Hom β€˜ndx), β„ŽβŸ©, ⟨(compβ€˜ndx), (π‘₯ ∈ (𝑏 Γ— 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd β€˜π‘₯)β„Žπ‘¦), 𝑓 ∈ (β„Žβ€˜π‘₯) ↦ ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩(compβ€˜π‘Ÿ)(1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩(compβ€˜π‘ )(2nd β€˜π‘¦))(2nd β€˜π‘“))⟩))⟩} ∈ V
32csbex 5302 . . 3 ⦋(𝑒 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1st β€˜π‘’)(Hom β€˜π‘Ÿ)(1st β€˜π‘£)) Γ— ((2nd β€˜π‘’)(Hom β€˜π‘ )(2nd β€˜π‘£)))) / β„Žβ¦Œ{⟨(Baseβ€˜ndx), π‘βŸ©, ⟨(Hom β€˜ndx), β„ŽβŸ©, ⟨(compβ€˜ndx), (π‘₯ ∈ (𝑏 Γ— 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd β€˜π‘₯)β„Žπ‘¦), 𝑓 ∈ (β„Žβ€˜π‘₯) ↦ ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩(compβ€˜π‘Ÿ)(1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩(compβ€˜π‘ )(2nd β€˜π‘¦))(2nd β€˜π‘“))⟩))⟩} ∈ V
43csbex 5302 . 2 ⦋((Baseβ€˜π‘Ÿ) Γ— (Baseβ€˜π‘ )) / π‘β¦Œβ¦‹(𝑒 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1st β€˜π‘’)(Hom β€˜π‘Ÿ)(1st β€˜π‘£)) Γ— ((2nd β€˜π‘’)(Hom β€˜π‘ )(2nd β€˜π‘£)))) / β„Žβ¦Œ{⟨(Baseβ€˜ndx), π‘βŸ©, ⟨(Hom β€˜ndx), β„ŽβŸ©, ⟨(compβ€˜ndx), (π‘₯ ∈ (𝑏 Γ— 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd β€˜π‘₯)β„Žπ‘¦), 𝑓 ∈ (β„Žβ€˜π‘₯) ↦ ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩(compβ€˜π‘Ÿ)(1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩(compβ€˜π‘ )(2nd β€˜π‘¦))(2nd β€˜π‘“))⟩))⟩} ∈ V
51, 4fnmpoi 8050 1 Γ—c Fn (V Γ— V)
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3466  β¦‹csb 3886  {ctp 4625  βŸ¨cop 4627   Γ— cxp 5665   Fn wfn 6529  β€˜cfv 6534  (class class class)co 7402   ∈ cmpo 7404  1st c1st 7967  2nd c2nd 7968  ndxcnx 17131  Basecbs 17149  Hom chom 17213  compcco 17214   Γ—c cxpc 18128
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-xpc 18132
This theorem is referenced by:  xpcbas  18138  xpchomfval  18139  xpccofval  18142
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