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Theorem xpcbas 18223
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 2737 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
5 eqid 2737 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
6 eqid 2737 . . . 4 (comp‘𝐶) = (comp‘𝐶)
7 eqid 2737 . . . 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 2738 . . . 4 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑋 × 𝑌) = (𝑋 × 𝑌))
11 eqidd 2738 . . . 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 2738 . . . 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 18222 . . 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 7773 . . . 4 (𝑋 × 𝑌) ∈ V
1716a1i 11 . . 3 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑋 × 𝑌) ∈ V)
1813, 17estrreslem1 18181 . 2 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑋 × 𝑌) = (Base‘𝑇))
19 base0 17252 . . 3 ∅ = (Base‘∅)
20 fvprc 6898 . . . . . 6 𝐶 ∈ V → (Base‘𝐶) = ∅)
212, 20eqtrid 2789 . . . . 5 𝐶 ∈ V → 𝑋 = ∅)
22 fvprc 6898 . . . . . 6 𝐷 ∈ V → (Base‘𝐷) = ∅)
233, 22eqtrid 2789 . . . . 5 𝐷 ∈ V → 𝑌 = ∅)
2421, 23orim12i 909 . . . 4 ((¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V) → (𝑋 = ∅ ∨ 𝑌 = ∅))
25 ianor 984 . . . 4 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V))
26 xpeq0 6180 . . . 4 ((𝑋 × 𝑌) = ∅ ↔ (𝑋 = ∅ ∨ 𝑌 = ∅))
2724, 25, 263imtr4i 292 . . 3 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑋 × 𝑌) = ∅)
28 fnxpc 18221 . . . . . . 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 7617 . . . . 5 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 ×c 𝐷) = ∅)
321, 31eqtrid 2789 . . . 4 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑇 = ∅)
3332fveq2d 6910 . . 3 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (Base‘𝑇) = (Base‘∅))
3419, 27, 333eqtr4a 2803 . 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 848   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333  cop 4632   × cxp 5683  dom cdm 5685   Fn wfn 6556  cfv 6561  (class class class)co 7431  cmpo 7433  1st c1st 8012  2nd c2nd 8013  Basecbs 17247  Hom chom 17308  compcco 17309   ×c cxpc 18213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-slot 17219  df-ndx 17231  df-base 17248  df-hom 17321  df-cco 17322  df-xpc 18217
This theorem is referenced by:  xpchomfval  18224  xpccofval  18227  xpchom2  18231  xpcco2  18232  xpccatid  18233  1stfval  18236  2ndfval  18239  1stfcl  18242  2ndfcl  18243  prfcl  18248  prf1st  18249  prf2nd  18250  1st2ndprf  18251  catcxpccl  18252  xpcpropd  18253  evlfcl  18267  curf1cl  18273  curf2cl  18276  curfcl  18277  uncf1  18281  uncf2  18282  uncfcurf  18284  diag11  18288  diag12  18289  diag2  18290  curf2ndf  18292  hofcl  18304  yonedalem21  18318  yonedalem22  18323  yonedalem3b  18324  yonedalem3  18325  yonedainv  18326  yonffthlem  18327  elxpcbasex1ALT  48955  elxpcbasex2ALT  48957  xpcfucbas  48958  dfswapf2  48967  swapf1a  48975  swapf1  48978  swapf2val  48979  swapf1f1o  48981  swapf2f1oa  48983  swapfida  48986  cofuswapf1  48994  cofuswapf2  48995  fucofulem2  49006
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