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Theorem xpccofval 17426
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 = (𝐶 ×c 𝐷)
xpccofval.b 𝐵 = (Base‘𝑇)
xpccofval.k 𝐾 = (Hom ‘𝑇)
xpccofval.o1 · = (comp‘𝐶)
xpccofval.o2 = (comp‘𝐷)
xpccofval.o 𝑂 = (comp‘𝑇)
Assertion
Ref Expression
xpccofval 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐾𝑦), 𝑓 ∈ (𝐾𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩))
Distinct variable groups:   𝑓,𝑔,𝑥,𝑦,𝐵   𝐶,𝑓,𝑔,𝑥,𝑦   𝐷,𝑓,𝑔,𝑥,𝑦   · ,𝑓,𝑔,𝑥,𝑦   ,𝑓,𝑔,𝑥,𝑦   𝑓,𝐾,𝑔,𝑥,𝑦   𝑥,𝑂,𝑦
Allowed substitution hints:   𝑇(𝑥,𝑦,𝑓,𝑔)   𝑂(𝑓,𝑔)

Proof of Theorem xpccofval
Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpccofval.t . . . 4 𝑇 = (𝐶 ×c 𝐷)
2 eqid 2821 . . . 4 (Base‘𝐶) = (Base‘𝐶)
3 eqid 2821 . . . 4 (Base‘𝐷) = (Base‘𝐷)
4 eqid 2821 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
5 eqid 2821 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
6 xpccofval.o1 . . . 4 · = (comp‘𝐶)
7 xpccofval.o2 . . . 4 = (comp‘𝐷)
8 simpl 485 . . . 4 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐶 ∈ V)
9 simpr 487 . . . 4 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐷 ∈ V)
10 xpccofval.b . . . . . 6 𝐵 = (Base‘𝑇)
111, 2, 3xpcbas 17422 . . . . . 6 ((Base‘𝐶) × (Base‘𝐷)) = (Base‘𝑇)
1210, 11eqtr4i 2847 . . . . 5 𝐵 = ((Base‘𝐶) × (Base‘𝐷))
1312a1i 11 . . . 4 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐵 = ((Base‘𝐶) × (Base‘𝐷)))
14 xpccofval.k . . . . . 6 𝐾 = (Hom ‘𝑇)
151, 10, 4, 5, 14xpchomfval 17423 . . . . 5 𝐾 = (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)(Hom ‘𝐶)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝐷)(2nd𝑣))))
1615a1i 11 . . . 4 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐾 = (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)(Hom ‘𝐶)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝐷)(2nd𝑣)))))
17 eqidd 2822 . . . 4 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐾𝑦), 𝑓 ∈ (𝐾𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩)) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐾𝑦), 𝑓 ∈ (𝐾𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩)))
181, 2, 3, 4, 5, 6, 7, 8, 9, 13, 16, 17xpcval 17421 . . 3 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑇 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐾𝑦), 𝑓 ∈ (𝐾𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩))⟩})
19 catstr 17221 . . 3 {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐾𝑦), 𝑓 ∈ (𝐾𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩))⟩} Struct ⟨1, 15⟩
20 ccoid 16684 . . 3 comp = Slot (comp‘ndx)
21 snsstp3 4744 . . 3 {⟨(comp‘ndx), (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐾𝑦), 𝑓 ∈ (𝐾𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩))⟩} ⊆ {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐾𝑦), 𝑓 ∈ (𝐾𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩))⟩}
2210fvexi 6678 . . . . . 6 𝐵 ∈ V
2322, 22xpex 7470 . . . . 5 (𝐵 × 𝐵) ∈ V
2423, 22mpoex 7771 . . . 4 (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐾𝑦), 𝑓 ∈ (𝐾𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩)) ∈ V
2524a1i 11 . . 3 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐾𝑦), 𝑓 ∈ (𝐾𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩)) ∈ V)
26 xpccofval.o . . 3 𝑂 = (comp‘𝑇)
2718, 19, 20, 21, 25, 26strfv3 16526 . 2 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐾𝑦), 𝑓 ∈ (𝐾𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩)))
28 fnxpc 17420 . . . . . . . 8 ×c Fn (V × V)
29 fndm 6449 . . . . . . . 8 ( ×c Fn (V × V) → dom ×c = (V × V))
3028, 29ax-mp 5 . . . . . . 7 dom ×c = (V × V)
3130ndmov 7326 . . . . . 6 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 ×c 𝐷) = ∅)
321, 31syl5eq 2868 . . . . 5 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑇 = ∅)
3332fveq2d 6668 . . . 4 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (comp‘𝑇) = (comp‘∅))
3420str0 16529 . . . 4 ∅ = (comp‘∅)
3533, 26, 343eqtr4g 2881 . . 3 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑂 = ∅)
3632fveq2d 6668 . . . . . 6 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (Base‘𝑇) = (Base‘∅))
37 base0 16530 . . . . . 6 ∅ = (Base‘∅)
3836, 10, 373eqtr4g 2881 . . . . 5 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐵 = ∅)
3938olcd 870 . . . 4 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → ((𝐵 × 𝐵) = ∅ ∨ 𝐵 = ∅))
40 0mpo0 7231 . . . 4 (((𝐵 × 𝐵) = ∅ ∨ 𝐵 = ∅) → (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐾𝑦), 𝑓 ∈ (𝐾𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩)) = ∅)
4139, 40syl 17 . . 3 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐾𝑦), 𝑓 ∈ (𝐾𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩)) = ∅)
4235, 41eqtr4d 2859 . 2 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐾𝑦), 𝑓 ∈ (𝐾𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩)))
4327, 42pm2.61i 184 1 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐾𝑦), 𝑓 ∈ (𝐾𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 398  wo 843   = wceq 1533  wcel 2110  Vcvv 3494  c0 4290  {ctp 4564  cop 4566   × cxp 5547  dom cdm 5549   Fn wfn 6344  cfv 6349  (class class class)co 7150  cmpo 7152  1st c1st 7681  2nd c2nd 7682  1c1 10532  5c5 11689  cdc 12092  ndxcnx 16474  Basecbs 16477  Hom chom 16570  compcco 16571   ×c cxpc 17412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8283  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-7 11699  df-8 11700  df-9 11701  df-n0 11892  df-z 11976  df-dec 12093  df-uz 12238  df-fz 12887  df-struct 16479  df-ndx 16480  df-slot 16481  df-base 16483  df-hom 16583  df-cco 16584  df-xpc 17416
This theorem is referenced by:  xpcco  17427
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