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Theorem xpchomfval 18067
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 2736 . . . 4 (Base‘𝐶) = (Base‘𝐶)
3 eqid 2736 . . . 4 (Base‘𝐷) = (Base‘𝐷)
4 xpchomfval.h . . . 4 𝐻 = (Hom ‘𝐶)
5 xpchomfval.j . . . 4 𝐽 = (Hom ‘𝐷)
6 eqid 2736 . . . 4 (comp‘𝐶) = (comp‘𝐶)
7 eqid 2736 . . . 4 (comp‘𝐷) = (comp‘𝐷)
8 simpl 483 . . . 4 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐶 ∈ V)
9 simpr 485 . . . 4 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐷 ∈ V)
10 xpchomfval.y . . . . . 6 𝐵 = (Base‘𝑇)
111, 2, 3xpcbas 18066 . . . . . 6 ((Base‘𝐶) × (Base‘𝐷)) = (Base‘𝑇)
1210, 11eqtr4i 2767 . . . . 5 𝐵 = ((Base‘𝐶) × (Base‘𝐷))
1312a1i 11 . . . 4 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐵 = ((Base‘𝐶) × (Base‘𝐷)))
14 eqidd 2737 . . . 4 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣)))) = (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣)))))
15 eqidd 2737 . . . 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 18065 . . 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 17845 . . 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 17293 . . 3 Hom = Slot (Hom ‘ndx)
19 snsstp2 4777 . . 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 6856 . . . . 5 𝐵 ∈ V
2120, 20mpoex 8012 . . . 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 17077 . 2 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐾 = (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣)))))
25 fnxpc 18064 . . . . . . . 8 ×c Fn (V × V)
26 fndm 6605 . . . . . . . 8 ( ×c Fn (V × V) → dom ×c = (V × V))
2725, 26ax-mp 5 . . . . . . 7 dom ×c = (V × V)
2827ndmov 7538 . . . . . 6 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 ×c 𝐷) = ∅)
291, 28eqtrid 2788 . . . . 5 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑇 = ∅)
3029fveq2d 6846 . . . 4 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (Hom ‘𝑇) = (Hom ‘∅))
3118str0 17061 . . . 4 ∅ = (Hom ‘∅)
3230, 23, 313eqtr4g 2801 . . 3 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐾 = ∅)
3329fveq2d 6846 . . . . . 6 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (Base‘𝑇) = (Base‘∅))
34 base0 17088 . . . . . 6 ∅ = (Base‘∅)
3533, 10, 343eqtr4g 2801 . . . . 5 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐵 = ∅)
3635olcd 872 . . . 4 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐵 = ∅ ∨ 𝐵 = ∅))
37 0mpo0 7440 . . . 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 2779 . 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 396  wo 845   = wceq 1541  wcel 2106  Vcvv 3445  c0 4282  {ctp 4590  cop 4592   × cxp 5631  dom cdm 5633   Fn wfn 6491  cfv 6496  (class class class)co 7357  cmpo 7359  1st c1st 7919  2nd c2nd 7920  1c1 11052  5c5 12211  cdc 12618  ndxcnx 17065  Basecbs 17083  Hom chom 17144  compcco 17145   ×c cxpc 18056
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-struct 17019  df-slot 17054  df-ndx 17066  df-base 17084  df-hom 17157  df-cco 17158  df-xpc 18060
This theorem is referenced by:  xpchom  18068  relxpchom  18069  xpccofval  18070  catcxpccl  18095  catcxpcclOLD  18096  xpcpropd  18097
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