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Theorem xpcbas 18233
Description: Set of objects of the binary product of categories. (Contributed by Mario Carneiro, 10-Jan-2017.)
Hypotheses
Ref Expression
xpcbas.t 𝑇 = (𝐶 ×c 𝐷)
xpcbas.x 𝑋 = (Base‘𝐶)
xpcbas.y 𝑌 = (Base‘𝐷)
Assertion
Ref Expression
xpcbas (𝑋 × 𝑌) = (Base‘𝑇)

Proof of Theorem xpcbas
Dummy variables 𝑓 𝑔 𝑢 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpcbas.t . . . 4 𝑇 = (𝐶 ×c 𝐷)
2 xpcbas.x . . . 4 𝑋 = (Base‘𝐶)
3 xpcbas.y . . . 4 𝑌 = (Base‘𝐷)
4 eqid 2734 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
5 eqid 2734 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
6 eqid 2734 . . . 4 (comp‘𝐶) = (comp‘𝐶)
7 eqid 2734 . . . 4 (comp‘𝐷) = (comp‘𝐷)
8 simpl 482 . . . 4 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐶 ∈ V)
9 simpr 484 . . . 4 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐷 ∈ V)
10 eqidd 2735 . . . 4 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑋 × 𝑌) = (𝑋 × 𝑌))
11 eqidd 2735 . . . 4 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ (((1st𝑢)(Hom ‘𝐶)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝐷)(2nd𝑣)))) = (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ (((1st𝑢)(Hom ‘𝐶)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝐷)(2nd𝑣)))))
12 eqidd 2735 . . . 4 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑥 ∈ ((𝑋 × 𝑌) × (𝑋 × 𝑌)), 𝑦 ∈ (𝑋 × 𝑌) ↦ (𝑔 ∈ ((2nd𝑥)(𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ (((1st𝑢)(Hom ‘𝐶)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝐷)(2nd𝑣))))𝑦), 𝑓 ∈ ((𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ (((1st𝑢)(Hom ‘𝐶)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝐷)(2nd𝑣))))‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝐶)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝐷)(2nd𝑦))(2nd𝑓))⟩)) = (𝑥 ∈ ((𝑋 × 𝑌) × (𝑋 × 𝑌)), 𝑦 ∈ (𝑋 × 𝑌) ↦ (𝑔 ∈ ((2nd𝑥)(𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ (((1st𝑢)(Hom ‘𝐶)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝐷)(2nd𝑣))))𝑦), 𝑓 ∈ ((𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ (((1st𝑢)(Hom ‘𝐶)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝐷)(2nd𝑣))))‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝐶)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝐷)(2nd𝑦))(2nd𝑓))⟩)))
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12xpcval 18232 . . 3 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑇 = {⟨(Base‘ndx), (𝑋 × 𝑌)⟩, ⟨(Hom ‘ndx), (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ (((1st𝑢)(Hom ‘𝐶)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝐷)(2nd𝑣))))⟩, ⟨(comp‘ndx), (𝑥 ∈ ((𝑋 × 𝑌) × (𝑋 × 𝑌)), 𝑦 ∈ (𝑋 × 𝑌) ↦ (𝑔 ∈ ((2nd𝑥)(𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ (((1st𝑢)(Hom ‘𝐶)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝐷)(2nd𝑣))))𝑦), 𝑓 ∈ ((𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ (((1st𝑢)(Hom ‘𝐶)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝐷)(2nd𝑣))))‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝐶)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝐷)(2nd𝑦))(2nd𝑓))⟩))⟩})
142fvexi 6920 . . . . 5 𝑋 ∈ V
153fvexi 6920 . . . . 5 𝑌 ∈ V
1614, 15xpex 7771 . . . 4 (𝑋 × 𝑌) ∈ V
1716a1i 11 . . 3 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑋 × 𝑌) ∈ V)
1813, 17estrreslem1 18191 . 2 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑋 × 𝑌) = (Base‘𝑇))
19 base0 17249 . . 3 ∅ = (Base‘∅)
20 fvprc 6898 . . . . . 6 𝐶 ∈ V → (Base‘𝐶) = ∅)
212, 20eqtrid 2786 . . . . 5 𝐶 ∈ V → 𝑋 = ∅)
22 fvprc 6898 . . . . . 6 𝐷 ∈ V → (Base‘𝐷) = ∅)
233, 22eqtrid 2786 . . . . 5 𝐷 ∈ V → 𝑌 = ∅)
2421, 23orim12i 908 . . . 4 ((¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V) → (𝑋 = ∅ ∨ 𝑌 = ∅))
25 ianor 983 . . . 4 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V))
26 xpeq0 6181 . . . 4 ((𝑋 × 𝑌) = ∅ ↔ (𝑋 = ∅ ∨ 𝑌 = ∅))
2724, 25, 263imtr4i 292 . . 3 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑋 × 𝑌) = ∅)
28 fnxpc 18231 . . . . . . 7 ×c Fn (V × V)
29 fndm 6671 . . . . . . 7 ( ×c Fn (V × V) → dom ×c = (V × V))
3028, 29ax-mp 5 . . . . . 6 dom ×c = (V × V)
3130ndmov 7616 . . . . 5 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 ×c 𝐷) = ∅)
321, 31eqtrid 2786 . . . 4 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑇 = ∅)
3332fveq2d 6910 . . 3 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (Base‘𝑇) = (Base‘∅))
3419, 27, 333eqtr4a 2800 . 2 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑋 × 𝑌) = (Base‘𝑇))
3518, 34pm2.61i 182 1 (𝑋 × 𝑌) = (Base‘𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395  wo 847   = wceq 1536  wcel 2105  Vcvv 3477  c0 4338  cop 4636   × cxp 5686  dom cdm 5688   Fn wfn 6557  cfv 6562  (class class class)co 7430  cmpo 7432  1st c1st 8010  2nd c2nd 8011  Basecbs 17244  Hom chom 17308  compcco 17309   ×c cxpc 18223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-slot 17215  df-ndx 17227  df-base 17245  df-hom 17321  df-cco 17322  df-xpc 18227
This theorem is referenced by:  xpchomfval  18234  xpccofval  18237  xpchom2  18241  xpcco2  18242  xpccatid  18243  1stfval  18246  2ndfval  18249  1stfcl  18252  2ndfcl  18253  prfcl  18258  prf1st  18259  prf2nd  18260  1st2ndprf  18261  catcxpccl  18262  catcxpcclOLD  18263  xpcpropd  18264  evlfcl  18278  curf1cl  18284  curf2cl  18287  curfcl  18288  uncf1  18292  uncf2  18293  uncfcurf  18295  diag11  18299  diag12  18300  diag2  18301  curf2ndf  18303  hofcl  18315  yonedalem21  18329  yonedalem22  18334  yonedalem3b  18335  yonedalem3  18336  yonedainv  18337  yonffthlem  18338
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