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Theorem fnxpc 18160
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 18156 . 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 7743 . . . 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 5305 . . 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 5305 . 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 8068 1 Γ—c Fn (V Γ— V)
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3470  β¦‹csb 3890  {ctp 4628  βŸ¨cop 4630   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   Γ—c cxpc 18152
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-xpc 18156
This theorem is referenced by:  xpcbas  18162  xpchomfval  18163  xpccofval  18166
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