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Theorem fnxpc 17414
Description: The binary product of categories is a two-argument function. (Contributed by Mario Carneiro, 10-Jan-2017.)
Assertion
Ref Expression
fnxpc ×c Fn (V × V)

Proof of Theorem fnxpc
Dummy variables 𝑓 𝑏 𝑔 𝑟 𝑠 𝑢 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-xpc 17410 . 2 ×c = (𝑟 ∈ V, 𝑠 ∈ V ↦ ((Base‘𝑟) × (Base‘𝑠)) / 𝑏(𝑢𝑏, 𝑣𝑏 ↦ (((1st𝑢)(Hom ‘𝑟)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑠)(2nd𝑣)))) / {⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦𝑏 ↦ (𝑔 ∈ ((2nd𝑥)𝑦), 𝑓 ∈ (𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑟)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑠)(2nd𝑦))(2nd𝑓))⟩))⟩})
2 tpex 7459 . . . 4 {⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦𝑏 ↦ (𝑔 ∈ ((2nd𝑥)𝑦), 𝑓 ∈ (𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑟)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑠)(2nd𝑦))(2nd𝑓))⟩))⟩} ∈ V
32csbex 5206 . . 3 (𝑢𝑏, 𝑣𝑏 ↦ (((1st𝑢)(Hom ‘𝑟)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑠)(2nd𝑣)))) / {⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦𝑏 ↦ (𝑔 ∈ ((2nd𝑥)𝑦), 𝑓 ∈ (𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑟)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑠)(2nd𝑦))(2nd𝑓))⟩))⟩} ∈ V
43csbex 5206 . 2 ((Base‘𝑟) × (Base‘𝑠)) / 𝑏(𝑢𝑏, 𝑣𝑏 ↦ (((1st𝑢)(Hom ‘𝑟)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑠)(2nd𝑣)))) / {⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦𝑏 ↦ (𝑔 ∈ ((2nd𝑥)𝑦), 𝑓 ∈ (𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑟)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑠)(2nd𝑦))(2nd𝑓))⟩))⟩} ∈ V
51, 4fnmpoi 7757 1 ×c Fn (V × V)
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3492  csb 3880  {ctp 4561  cop 4563   × cxp 5546   Fn wfn 6343  cfv 6348  (class class class)co 7145  cmpo 7147  1st c1st 7676  2nd c2nd 7677  ndxcnx 16468  Basecbs 16471  Hom chom 16564  compcco 16565   ×c cxpc 17406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-xpc 17410
This theorem is referenced by:  xpcbas  17416  xpchomfval  17417  xpccofval  17420
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