 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpchomfval Structured version   Visualization version   GIF version

Theorem xpchomfval 17012
 Description: Set of morphisms of the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
xpchomfval.t 𝑇 = (𝐶 ×c 𝐷)
xpchomfval.y 𝐵 = (Base‘𝑇)
xpchomfval.h 𝐻 = (Hom ‘𝐶)
xpchomfval.j 𝐽 = (Hom ‘𝐷)
xpchomfval.k 𝐾 = (Hom ‘𝑇)
Assertion
Ref Expression
xpchomfval 𝐾 = (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣))))
Distinct variable groups:   𝑣,𝑢,𝐵   𝑢,𝐶,𝑣   𝑢,𝐷,𝑣   𝑢,𝐻,𝑣   𝑢,𝐽,𝑣
Allowed substitution hints:   𝑇(𝑣,𝑢)   𝐾(𝑣,𝑢)

Proof of Theorem xpchomfval
Dummy variables 𝑓 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpchomfval.t . . . 4 𝑇 = (𝐶 ×c 𝐷)
2 eqid 2752 . . . 4 (Base‘𝐶) = (Base‘𝐶)
3 eqid 2752 . . . 4 (Base‘𝐷) = (Base‘𝐷)
4 xpchomfval.h . . . 4 𝐻 = (Hom ‘𝐶)
5 xpchomfval.j . . . 4 𝐽 = (Hom ‘𝐷)
6 eqid 2752 . . . 4 (comp‘𝐶) = (comp‘𝐶)
7 eqid 2752 . . . 4 (comp‘𝐷) = (comp‘𝐷)
8 simpl 474 . . . 4 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐶 ∈ V)
9 simpr 479 . . . 4 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐷 ∈ V)
10 xpchomfval.y . . . . . 6 𝐵 = (Base‘𝑇)
111, 2, 3xpcbas 17011 . . . . . 6 ((Base‘𝐶) × (Base‘𝐷)) = (Base‘𝑇)
1210, 11eqtr4i 2777 . . . . 5 𝐵 = ((Base‘𝐶) × (Base‘𝐷))
1312a1i 11 . . . 4 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐵 = ((Base‘𝐶) × (Base‘𝐷)))
14 eqidd 2753 . . . 4 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣)))) = (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣)))))
15 eqidd 2753 . . . 4 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)(𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣))))𝑦), 𝑓 ∈ ((𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣))))‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝐶)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝐷)(2nd𝑦))(2nd𝑓))⟩)) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)(𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣))))𝑦), 𝑓 ∈ ((𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣))))‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝐶)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝐷)(2nd𝑦))(2nd𝑓))⟩)))
161, 2, 3, 4, 5, 6, 7, 8, 9, 13, 14, 15xpcval 17010 . . 3 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑇 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣))))⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)(𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣))))𝑦), 𝑓 ∈ ((𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣))))‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝐶)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝐷)(2nd𝑦))(2nd𝑓))⟩))⟩})
17 catstr 16810 . . 3 {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣))))⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)(𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣))))𝑦), 𝑓 ∈ ((𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣))))‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝐶)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝐷)(2nd𝑦))(2nd𝑓))⟩))⟩} Struct ⟨1, 15⟩
18 homid 16269 . . 3 Hom = Slot (Hom ‘ndx)
19 snsstp2 4485 . . 3 {⟨(Hom ‘ndx), (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣))))⟩} ⊆ {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣))))⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)(𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣))))𝑦), 𝑓 ∈ ((𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣))))‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝐶)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝐷)(2nd𝑦))(2nd𝑓))⟩))⟩}
20 fvex 6354 . . . . . 6 (Base‘𝑇) ∈ V
2110, 20eqeltri 2827 . . . . 5 𝐵 ∈ V
2221, 21mpt2ex 7407 . . . 4 (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣)))) ∈ V
2322a1i 11 . . 3 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣)))) ∈ V)
24 xpchomfval.k . . 3 𝐾 = (Hom ‘𝑇)
2516, 17, 18, 19, 23, 24strfv3 16102 . 2 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐾 = (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣)))))
26 mpt20 6882 . . . 4 (𝑢 ∈ ∅, 𝑣 ∈ ∅ ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣)))) = ∅
2726eqcomi 2761 . . 3 ∅ = (𝑢 ∈ ∅, 𝑣 ∈ ∅ ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣))))
28 fnxpc 17009 . . . . . . . 8 ×c Fn (V × V)
29 fndm 6143 . . . . . . . 8 ( ×c Fn (V × V) → dom ×c = (V × V))
3028, 29ax-mp 5 . . . . . . 7 dom ×c = (V × V)
3130ndmov 6975 . . . . . 6 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 ×c 𝐷) = ∅)
321, 31syl5eq 2798 . . . . 5 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑇 = ∅)
3332fveq2d 6348 . . . 4 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (Hom ‘𝑇) = (Hom ‘∅))
3418str0 16105 . . . 4 ∅ = (Hom ‘∅)
3533, 24, 343eqtr4g 2811 . . 3 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐾 = ∅)
3632fveq2d 6348 . . . . 5 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (Base‘𝑇) = (Base‘∅))
37 base0 16106 . . . . 5 ∅ = (Base‘∅)
3836, 10, 373eqtr4g 2811 . . . 4 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐵 = ∅)
39 eqidd 2753 . . . 4 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣))) = (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣))))
4038, 38, 39mpt2eq123dv 6874 . . 3 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣)))) = (𝑢 ∈ ∅, 𝑣 ∈ ∅ ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣)))))
4127, 35, 403eqtr4a 2812 . 2 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐾 = (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣)))))
4225, 41pm2.61i 176 1 𝐾 = (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 383   = wceq 1624   ∈ wcel 2131  Vcvv 3332  ∅c0 4050  {ctp 4317  ⟨cop 4319   × cxp 5256  dom cdm 5258   Fn wfn 6036  ‘cfv 6041  (class class class)co 6805   ↦ cmpt2 6807  1st c1st 7323  2nd c2nd 7324  1c1 10121  5c5 11257  ;cdc 11677  ndxcnx 16048  Basecbs 16051  Hom chom 16146  compcco 16147   ×c cxpc 17001 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-fal 1630  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-oadd 7725  df-er 7903  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-nn 11205  df-2 11263  df-3 11264  df-4 11265  df-5 11266  df-6 11267  df-7 11268  df-8 11269  df-9 11270  df-n0 11477  df-z 11562  df-dec 11678  df-uz 11872  df-fz 12512  df-struct 16053  df-ndx 16054  df-slot 16055  df-base 16057  df-hom 16160  df-cco 16161  df-xpc 17005 This theorem is referenced by:  xpchom  17013  relxpchom  17014  xpccofval  17015  catcxpccl  17040  xpcpropd  17041
 Copyright terms: Public domain W3C validator