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Theorem xpchomfval 17994
Description: Set of morphisms of the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.) (Proof shortened by AV, 1-Mar-2024.)
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 2737 . . . 4 (Base‘𝐶) = (Base‘𝐶)
3 eqid 2737 . . . 4 (Base‘𝐷) = (Base‘𝐷)
4 xpchomfval.h . . . 4 𝐻 = (Hom ‘𝐶)
5 xpchomfval.j . . . 4 𝐽 = (Hom ‘𝐷)
6 eqid 2737 . . . 4 (comp‘𝐶) = (comp‘𝐶)
7 eqid 2737 . . . 4 (comp‘𝐷) = (comp‘𝐷)
8 simpl 484 . . . 4 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐶 ∈ V)
9 simpr 486 . . . 4 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐷 ∈ V)
10 xpchomfval.y . . . . . 6 𝐵 = (Base‘𝑇)
111, 2, 3xpcbas 17993 . . . . . 6 ((Base‘𝐶) × (Base‘𝐷)) = (Base‘𝑇)
1210, 11eqtr4i 2768 . . . . 5 𝐵 = ((Base‘𝐶) × (Base‘𝐷))
1312a1i 11 . . . 4 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐵 = ((Base‘𝐶) × (Base‘𝐷)))
14 eqidd 2738 . . . 4 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣)))) = (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣)))))
15 eqidd 2738 . . . 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 17992 . . 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 17772 . . 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 17220 . . 3 Hom = Slot (Hom ‘ndx)
19 snsstp2 4769 . . 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))⟩))⟩}
2010fvexi 6844 . . . . 5 𝐵 ∈ V
2120, 20mpoex 7993 . . . 4 (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣)))) ∈ V
2221a1i 11 . . 3 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣)))) ∈ V)
23 xpchomfval.k . . 3 𝐾 = (Hom ‘𝑇)
2416, 17, 18, 19, 22, 23strfv3 17004 . 2 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐾 = (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣)))))
25 fnxpc 17991 . . . . . . . 8 ×c Fn (V × V)
26 fndm 6593 . . . . . . . 8 ( ×c Fn (V × V) → dom ×c = (V × V))
2725, 26ax-mp 5 . . . . . . 7 dom ×c = (V × V)
2827ndmov 7523 . . . . . 6 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 ×c 𝐷) = ∅)
291, 28eqtrid 2789 . . . . 5 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑇 = ∅)
3029fveq2d 6834 . . . 4 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (Hom ‘𝑇) = (Hom ‘∅))
3118str0 16988 . . . 4 ∅ = (Hom ‘∅)
3230, 23, 313eqtr4g 2802 . . 3 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐾 = ∅)
3329fveq2d 6834 . . . . . 6 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (Base‘𝑇) = (Base‘∅))
34 base0 17015 . . . . . 6 ∅ = (Base‘∅)
3533, 10, 343eqtr4g 2802 . . . . 5 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐵 = ∅)
3635olcd 872 . . . 4 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐵 = ∅ ∨ 𝐵 = ∅))
37 0mpo0 7425 . . . 4 ((𝐵 = ∅ ∨ 𝐵 = ∅) → (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣)))) = ∅)
3836, 37syl 17 . . 3 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣)))) = ∅)
3932, 38eqtr4d 2780 . 2 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐾 = (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣)))))
4024, 39pm2.61i 182 1 𝐾 = (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 397  wo 845   = wceq 1541  wcel 2106  Vcvv 3442  c0 4274  {ctp 4582  cop 4584   × cxp 5623  dom cdm 5625   Fn wfn 6479  cfv 6484  (class class class)co 7342  cmpo 7344  1st c1st 7902  2nd c2nd 7903  1c1 10978  5c5 12137  cdc 12543  ndxcnx 16992  Basecbs 17010  Hom chom 17071  compcco 17072   ×c cxpc 17983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5234  ax-sep 5248  ax-nul 5255  ax-pow 5313  ax-pr 5377  ax-un 7655  ax-cnex 11033  ax-resscn 11034  ax-1cn 11035  ax-icn 11036  ax-addcl 11037  ax-addrcl 11038  ax-mulcl 11039  ax-mulrcl 11040  ax-mulcom 11041  ax-addass 11042  ax-mulass 11043  ax-distr 11044  ax-i2m1 11045  ax-1ne0 11046  ax-1rid 11047  ax-rnegex 11048  ax-rrecex 11049  ax-cnre 11050  ax-pre-lttri 11051  ax-pre-lttrn 11052  ax-pre-ltadd 11053  ax-pre-mulgt0 11054
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3732  df-csb 3848  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3921  df-nul 4275  df-if 4479  df-pw 4554  df-sn 4579  df-pr 4581  df-tp 4583  df-op 4585  df-uni 4858  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5181  df-tr 5215  df-id 5523  df-eprel 5529  df-po 5537  df-so 5538  df-fr 5580  df-we 5582  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6243  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6436  df-fun 6486  df-fn 6487  df-f 6488  df-f1 6489  df-fo 6490  df-f1o 6491  df-fv 6492  df-riota 7298  df-ov 7345  df-oprab 7346  df-mpo 7347  df-om 7786  df-1st 7904  df-2nd 7905  df-frecs 8172  df-wrecs 8203  df-recs 8277  df-rdg 8316  df-1o 8372  df-er 8574  df-en 8810  df-dom 8811  df-sdom 8812  df-fin 8813  df-pnf 11117  df-mnf 11118  df-xr 11119  df-ltxr 11120  df-le 11121  df-sub 11313  df-neg 11314  df-nn 12080  df-2 12142  df-3 12143  df-4 12144  df-5 12145  df-6 12146  df-7 12147  df-8 12148  df-9 12149  df-n0 12340  df-z 12426  df-dec 12544  df-uz 12689  df-fz 13346  df-struct 16946  df-slot 16981  df-ndx 16993  df-base 17011  df-hom 17084  df-cco 17085  df-xpc 17987
This theorem is referenced by:  xpchom  17995  relxpchom  17996  xpccofval  17997  catcxpccl  18022  catcxpcclOLD  18023  xpcpropd  18024
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