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Theorem xpccofval 18172
Description: Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.) (Proof shortened by AV, 2-Mar-2024.)
Hypotheses
Ref Expression
xpccofval.t 𝑇 = (𝐢 Γ—c 𝐷)
xpccofval.b 𝐡 = (Baseβ€˜π‘‡)
xpccofval.k 𝐾 = (Hom β€˜π‘‡)
xpccofval.o1 Β· = (compβ€˜πΆ)
xpccofval.o2 βˆ™ = (compβ€˜π·)
xpccofval.o 𝑂 = (compβ€˜π‘‡)
Assertion
Ref Expression
xpccofval 𝑂 = (π‘₯ ∈ (𝐡 Γ— 𝐡), 𝑦 ∈ 𝐡 ↦ (𝑔 ∈ ((2nd β€˜π‘₯)𝐾𝑦), 𝑓 ∈ (πΎβ€˜π‘₯) ↦ ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩ Β· (1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩ βˆ™ (2nd β€˜π‘¦))(2nd β€˜π‘“))⟩))
Distinct variable groups:   𝑓,𝑔,π‘₯,𝑦,𝐡   𝐢,𝑓,𝑔,π‘₯,𝑦   𝐷,𝑓,𝑔,π‘₯,𝑦   Β· ,𝑓,𝑔,π‘₯,𝑦   βˆ™ ,𝑓,𝑔,π‘₯,𝑦   𝑓,𝐾,𝑔,π‘₯,𝑦   π‘₯,𝑂,𝑦
Allowed substitution hints:   𝑇(π‘₯,𝑦,𝑓,𝑔)   𝑂(𝑓,𝑔)

Proof of Theorem xpccofval
Dummy variables 𝑒 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpccofval.t . . . 4 𝑇 = (𝐢 Γ—c 𝐷)
2 eqid 2725 . . . 4 (Baseβ€˜πΆ) = (Baseβ€˜πΆ)
3 eqid 2725 . . . 4 (Baseβ€˜π·) = (Baseβ€˜π·)
4 eqid 2725 . . . 4 (Hom β€˜πΆ) = (Hom β€˜πΆ)
5 eqid 2725 . . . 4 (Hom β€˜π·) = (Hom β€˜π·)
6 xpccofval.o1 . . . 4 Β· = (compβ€˜πΆ)
7 xpccofval.o2 . . . 4 βˆ™ = (compβ€˜π·)
8 simpl 481 . . . 4 ((𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ 𝐢 ∈ V)
9 simpr 483 . . . 4 ((𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ 𝐷 ∈ V)
10 xpccofval.b . . . . . 6 𝐡 = (Baseβ€˜π‘‡)
111, 2, 3xpcbas 18168 . . . . . 6 ((Baseβ€˜πΆ) Γ— (Baseβ€˜π·)) = (Baseβ€˜π‘‡)
1210, 11eqtr4i 2756 . . . . 5 𝐡 = ((Baseβ€˜πΆ) Γ— (Baseβ€˜π·))
1312a1i 11 . . . 4 ((𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ 𝐡 = ((Baseβ€˜πΆ) Γ— (Baseβ€˜π·)))
14 xpccofval.k . . . . . 6 𝐾 = (Hom β€˜π‘‡)
151, 10, 4, 5, 14xpchomfval 18169 . . . . 5 𝐾 = (𝑒 ∈ 𝐡, 𝑣 ∈ 𝐡 ↦ (((1st β€˜π‘’)(Hom β€˜πΆ)(1st β€˜π‘£)) Γ— ((2nd β€˜π‘’)(Hom β€˜π·)(2nd β€˜π‘£))))
1615a1i 11 . . . 4 ((𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ 𝐾 = (𝑒 ∈ 𝐡, 𝑣 ∈ 𝐡 ↦ (((1st β€˜π‘’)(Hom β€˜πΆ)(1st β€˜π‘£)) Γ— ((2nd β€˜π‘’)(Hom β€˜π·)(2nd β€˜π‘£)))))
17 eqidd 2726 . . . 4 ((𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ (π‘₯ ∈ (𝐡 Γ— 𝐡), 𝑦 ∈ 𝐡 ↦ (𝑔 ∈ ((2nd β€˜π‘₯)𝐾𝑦), 𝑓 ∈ (πΎβ€˜π‘₯) ↦ ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩ Β· (1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩ βˆ™ (2nd β€˜π‘¦))(2nd β€˜π‘“))⟩)) = (π‘₯ ∈ (𝐡 Γ— 𝐡), 𝑦 ∈ 𝐡 ↦ (𝑔 ∈ ((2nd β€˜π‘₯)𝐾𝑦), 𝑓 ∈ (πΎβ€˜π‘₯) ↦ ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩ Β· (1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩ βˆ™ (2nd β€˜π‘¦))(2nd β€˜π‘“))⟩)))
181, 2, 3, 4, 5, 6, 7, 8, 9, 13, 16, 17xpcval 18167 . . 3 ((𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ 𝑇 = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(Hom β€˜ndx), 𝐾⟩, ⟨(compβ€˜ndx), (π‘₯ ∈ (𝐡 Γ— 𝐡), 𝑦 ∈ 𝐡 ↦ (𝑔 ∈ ((2nd β€˜π‘₯)𝐾𝑦), 𝑓 ∈ (πΎβ€˜π‘₯) ↦ ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩ Β· (1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩ βˆ™ (2nd β€˜π‘¦))(2nd β€˜π‘“))⟩))⟩})
19 catstr 17947 . . 3 {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(Hom β€˜ndx), 𝐾⟩, ⟨(compβ€˜ndx), (π‘₯ ∈ (𝐡 Γ— 𝐡), 𝑦 ∈ 𝐡 ↦ (𝑔 ∈ ((2nd β€˜π‘₯)𝐾𝑦), 𝑓 ∈ (πΎβ€˜π‘₯) ↦ ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩ Β· (1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩ βˆ™ (2nd β€˜π‘¦))(2nd β€˜π‘“))⟩))⟩} Struct ⟨1, 15⟩
20 ccoid 17394 . . 3 comp = Slot (compβ€˜ndx)
21 snsstp3 4817 . . 3 {⟨(compβ€˜ndx), (π‘₯ ∈ (𝐡 Γ— 𝐡), 𝑦 ∈ 𝐡 ↦ (𝑔 ∈ ((2nd β€˜π‘₯)𝐾𝑦), 𝑓 ∈ (πΎβ€˜π‘₯) ↦ ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩ Β· (1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩ βˆ™ (2nd β€˜π‘¦))(2nd β€˜π‘“))⟩))⟩} βŠ† {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(Hom β€˜ndx), 𝐾⟩, ⟨(compβ€˜ndx), (π‘₯ ∈ (𝐡 Γ— 𝐡), 𝑦 ∈ 𝐡 ↦ (𝑔 ∈ ((2nd β€˜π‘₯)𝐾𝑦), 𝑓 ∈ (πΎβ€˜π‘₯) ↦ ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩ Β· (1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩ βˆ™ (2nd β€˜π‘¦))(2nd β€˜π‘“))⟩))⟩}
2210fvexi 6906 . . . . . 6 𝐡 ∈ V
2322, 22xpex 7753 . . . . 5 (𝐡 Γ— 𝐡) ∈ V
2423, 22mpoex 8082 . . . 4 (π‘₯ ∈ (𝐡 Γ— 𝐡), 𝑦 ∈ 𝐡 ↦ (𝑔 ∈ ((2nd β€˜π‘₯)𝐾𝑦), 𝑓 ∈ (πΎβ€˜π‘₯) ↦ ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩ Β· (1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩ βˆ™ (2nd β€˜π‘¦))(2nd β€˜π‘“))⟩)) ∈ V
2524a1i 11 . . 3 ((𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ (π‘₯ ∈ (𝐡 Γ— 𝐡), 𝑦 ∈ 𝐡 ↦ (𝑔 ∈ ((2nd β€˜π‘₯)𝐾𝑦), 𝑓 ∈ (πΎβ€˜π‘₯) ↦ ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩ Β· (1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩ βˆ™ (2nd β€˜π‘¦))(2nd β€˜π‘“))⟩)) ∈ V)
26 xpccofval.o . . 3 𝑂 = (compβ€˜π‘‡)
2718, 19, 20, 21, 25, 26strfv3 17173 . 2 ((𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ 𝑂 = (π‘₯ ∈ (𝐡 Γ— 𝐡), 𝑦 ∈ 𝐡 ↦ (𝑔 ∈ ((2nd β€˜π‘₯)𝐾𝑦), 𝑓 ∈ (πΎβ€˜π‘₯) ↦ ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩ Β· (1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩ βˆ™ (2nd β€˜π‘¦))(2nd β€˜π‘“))⟩)))
28 fnxpc 18166 . . . . . . . 8 Γ—c Fn (V Γ— V)
2928fndmi 6653 . . . . . . 7 dom Γ—c = (V Γ— V)
3029ndmov 7602 . . . . . 6 (Β¬ (𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ (𝐢 Γ—c 𝐷) = βˆ…)
311, 30eqtrid 2777 . . . . 5 (Β¬ (𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ 𝑇 = βˆ…)
3231fveq2d 6896 . . . 4 (Β¬ (𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ (compβ€˜π‘‡) = (compβ€˜βˆ…))
3320str0 17157 . . . 4 βˆ… = (compβ€˜βˆ…)
3432, 26, 333eqtr4g 2790 . . 3 (Β¬ (𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ 𝑂 = βˆ…)
3531fveq2d 6896 . . . . . 6 (Β¬ (𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ (Baseβ€˜π‘‡) = (Baseβ€˜βˆ…))
36 base0 17184 . . . . . 6 βˆ… = (Baseβ€˜βˆ…)
3735, 10, 363eqtr4g 2790 . . . . 5 (Β¬ (𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ 𝐡 = βˆ…)
3837olcd 872 . . . 4 (Β¬ (𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ ((𝐡 Γ— 𝐡) = βˆ… ∨ 𝐡 = βˆ…))
39 0mpo0 7500 . . . 4 (((𝐡 Γ— 𝐡) = βˆ… ∨ 𝐡 = βˆ…) β†’ (π‘₯ ∈ (𝐡 Γ— 𝐡), 𝑦 ∈ 𝐡 ↦ (𝑔 ∈ ((2nd β€˜π‘₯)𝐾𝑦), 𝑓 ∈ (πΎβ€˜π‘₯) ↦ ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩ Β· (1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩ βˆ™ (2nd β€˜π‘¦))(2nd β€˜π‘“))⟩)) = βˆ…)
4038, 39syl 17 . . 3 (Β¬ (𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ (π‘₯ ∈ (𝐡 Γ— 𝐡), 𝑦 ∈ 𝐡 ↦ (𝑔 ∈ ((2nd β€˜π‘₯)𝐾𝑦), 𝑓 ∈ (πΎβ€˜π‘₯) ↦ ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩ Β· (1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩ βˆ™ (2nd β€˜π‘¦))(2nd β€˜π‘“))⟩)) = βˆ…)
4134, 40eqtr4d 2768 . 2 (Β¬ (𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ 𝑂 = (π‘₯ ∈ (𝐡 Γ— 𝐡), 𝑦 ∈ 𝐡 ↦ (𝑔 ∈ ((2nd β€˜π‘₯)𝐾𝑦), 𝑓 ∈ (πΎβ€˜π‘₯) ↦ ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩ Β· (1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩ βˆ™ (2nd β€˜π‘¦))(2nd β€˜π‘“))⟩)))
4227, 41pm2.61i 182 1 𝑂 = (π‘₯ ∈ (𝐡 Γ— 𝐡), 𝑦 ∈ 𝐡 ↦ (𝑔 ∈ ((2nd β€˜π‘₯)𝐾𝑦), 𝑓 ∈ (πΎβ€˜π‘₯) ↦ ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩ Β· (1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩ βˆ™ (2nd β€˜π‘¦))(2nd β€˜π‘“))⟩))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   ∧ wa 394   ∨ wo 845   = wceq 1533   ∈ wcel 2098  Vcvv 3463  βˆ…c0 4318  {ctp 4628  βŸ¨cop 4630   Γ— cxp 5670  β€˜cfv 6543  (class class class)co 7416   ∈ cmpo 7418  1st c1st 7989  2nd c2nd 7990  1c1 11139  5c5 12300  cdc 12707  ndxcnx 17161  Basecbs 17179  Hom chom 17243  compcco 17244   Γ—c cxpc 18158
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 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  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-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7991  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8723  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-nn 12243  df-2 12305  df-3 12306  df-4 12307  df-5 12308  df-6 12309  df-7 12310  df-8 12311  df-9 12312  df-n0 12503  df-z 12589  df-dec 12708  df-uz 12853  df-fz 13517  df-struct 17115  df-slot 17150  df-ndx 17162  df-base 17180  df-hom 17256  df-cco 17257  df-xpc 18162
This theorem is referenced by:  xpcco  18173
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