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Theorem xpcbas 18126
Description: Set of objects of the binary product of categories. (Contributed by Mario Carneiro, 10-Jan-2017.)
Hypotheses
Ref Expression
xpcbas.t 𝑇 = (𝐢 Γ—c 𝐷)
xpcbas.x 𝑋 = (Baseβ€˜πΆ)
xpcbas.y π‘Œ = (Baseβ€˜π·)
Assertion
Ref Expression
xpcbas (𝑋 Γ— π‘Œ) = (Baseβ€˜π‘‡)

Proof of Theorem xpcbas
Dummy variables 𝑓 𝑔 𝑒 𝑣 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpcbas.t . . . 4 𝑇 = (𝐢 Γ—c 𝐷)
2 xpcbas.x . . . 4 𝑋 = (Baseβ€˜πΆ)
3 xpcbas.y . . . 4 π‘Œ = (Baseβ€˜π·)
4 eqid 2732 . . . 4 (Hom β€˜πΆ) = (Hom β€˜πΆ)
5 eqid 2732 . . . 4 (Hom β€˜π·) = (Hom β€˜π·)
6 eqid 2732 . . . 4 (compβ€˜πΆ) = (compβ€˜πΆ)
7 eqid 2732 . . . 4 (compβ€˜π·) = (compβ€˜π·)
8 simpl 483 . . . 4 ((𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ 𝐢 ∈ V)
9 simpr 485 . . . 4 ((𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ 𝐷 ∈ V)
10 eqidd 2733 . . . 4 ((𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ (𝑋 Γ— π‘Œ) = (𝑋 Γ— π‘Œ))
11 eqidd 2733 . . . 4 ((𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ (𝑒 ∈ (𝑋 Γ— π‘Œ), 𝑣 ∈ (𝑋 Γ— π‘Œ) ↦ (((1st β€˜π‘’)(Hom β€˜πΆ)(1st β€˜π‘£)) Γ— ((2nd β€˜π‘’)(Hom β€˜π·)(2nd β€˜π‘£)))) = (𝑒 ∈ (𝑋 Γ— π‘Œ), 𝑣 ∈ (𝑋 Γ— π‘Œ) ↦ (((1st β€˜π‘’)(Hom β€˜πΆ)(1st β€˜π‘£)) Γ— ((2nd β€˜π‘’)(Hom β€˜π·)(2nd β€˜π‘£)))))
12 eqidd 2733 . . . 4 ((𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ (π‘₯ ∈ ((𝑋 Γ— π‘Œ) Γ— (𝑋 Γ— π‘Œ)), 𝑦 ∈ (𝑋 Γ— π‘Œ) ↦ (𝑔 ∈ ((2nd β€˜π‘₯)(𝑒 ∈ (𝑋 Γ— π‘Œ), 𝑣 ∈ (𝑋 Γ— π‘Œ) ↦ (((1st β€˜π‘’)(Hom β€˜πΆ)(1st β€˜π‘£)) Γ— ((2nd β€˜π‘’)(Hom β€˜π·)(2nd β€˜π‘£))))𝑦), 𝑓 ∈ ((𝑒 ∈ (𝑋 Γ— π‘Œ), 𝑣 ∈ (𝑋 Γ— π‘Œ) ↦ (((1st β€˜π‘’)(Hom β€˜πΆ)(1st β€˜π‘£)) Γ— ((2nd β€˜π‘’)(Hom β€˜π·)(2nd β€˜π‘£))))β€˜π‘₯) ↦ ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩(compβ€˜πΆ)(1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩(compβ€˜π·)(2nd β€˜π‘¦))(2nd β€˜π‘“))⟩)) = (π‘₯ ∈ ((𝑋 Γ— π‘Œ) Γ— (𝑋 Γ— π‘Œ)), 𝑦 ∈ (𝑋 Γ— π‘Œ) ↦ (𝑔 ∈ ((2nd β€˜π‘₯)(𝑒 ∈ (𝑋 Γ— π‘Œ), 𝑣 ∈ (𝑋 Γ— π‘Œ) ↦ (((1st β€˜π‘’)(Hom β€˜πΆ)(1st β€˜π‘£)) Γ— ((2nd β€˜π‘’)(Hom β€˜π·)(2nd β€˜π‘£))))𝑦), 𝑓 ∈ ((𝑒 ∈ (𝑋 Γ— π‘Œ), 𝑣 ∈ (𝑋 Γ— π‘Œ) ↦ (((1st β€˜π‘’)(Hom β€˜πΆ)(1st β€˜π‘£)) Γ— ((2nd β€˜π‘’)(Hom β€˜π·)(2nd β€˜π‘£))))β€˜π‘₯) ↦ ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩(compβ€˜πΆ)(1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩(compβ€˜π·)(2nd β€˜π‘¦))(2nd β€˜π‘“))⟩)))
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12xpcval 18125 . . 3 ((𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ 𝑇 = {⟨(Baseβ€˜ndx), (𝑋 Γ— π‘Œ)⟩, ⟨(Hom β€˜ndx), (𝑒 ∈ (𝑋 Γ— π‘Œ), 𝑣 ∈ (𝑋 Γ— π‘Œ) ↦ (((1st β€˜π‘’)(Hom β€˜πΆ)(1st β€˜π‘£)) Γ— ((2nd β€˜π‘’)(Hom β€˜π·)(2nd β€˜π‘£))))⟩, ⟨(compβ€˜ndx), (π‘₯ ∈ ((𝑋 Γ— π‘Œ) Γ— (𝑋 Γ— π‘Œ)), 𝑦 ∈ (𝑋 Γ— π‘Œ) ↦ (𝑔 ∈ ((2nd β€˜π‘₯)(𝑒 ∈ (𝑋 Γ— π‘Œ), 𝑣 ∈ (𝑋 Γ— π‘Œ) ↦ (((1st β€˜π‘’)(Hom β€˜πΆ)(1st β€˜π‘£)) Γ— ((2nd β€˜π‘’)(Hom β€˜π·)(2nd β€˜π‘£))))𝑦), 𝑓 ∈ ((𝑒 ∈ (𝑋 Γ— π‘Œ), 𝑣 ∈ (𝑋 Γ— π‘Œ) ↦ (((1st β€˜π‘’)(Hom β€˜πΆ)(1st β€˜π‘£)) Γ— ((2nd β€˜π‘’)(Hom β€˜π·)(2nd β€˜π‘£))))β€˜π‘₯) ↦ ⟨((1st β€˜π‘”)(⟨(1st β€˜(1st β€˜π‘₯)), (1st β€˜(2nd β€˜π‘₯))⟩(compβ€˜πΆ)(1st β€˜π‘¦))(1st β€˜π‘“)), ((2nd β€˜π‘”)(⟨(2nd β€˜(1st β€˜π‘₯)), (2nd β€˜(2nd β€˜π‘₯))⟩(compβ€˜π·)(2nd β€˜π‘¦))(2nd β€˜π‘“))⟩))⟩})
142fvexi 6902 . . . . 5 𝑋 ∈ V
153fvexi 6902 . . . . 5 π‘Œ ∈ V
1614, 15xpex 7736 . . . 4 (𝑋 Γ— π‘Œ) ∈ V
1716a1i 11 . . 3 ((𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ (𝑋 Γ— π‘Œ) ∈ V)
1813, 17estrreslem1 18084 . 2 ((𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ (𝑋 Γ— π‘Œ) = (Baseβ€˜π‘‡))
19 base0 17145 . . 3 βˆ… = (Baseβ€˜βˆ…)
20 fvprc 6880 . . . . . 6 (Β¬ 𝐢 ∈ V β†’ (Baseβ€˜πΆ) = βˆ…)
212, 20eqtrid 2784 . . . . 5 (Β¬ 𝐢 ∈ V β†’ 𝑋 = βˆ…)
22 fvprc 6880 . . . . . 6 (Β¬ 𝐷 ∈ V β†’ (Baseβ€˜π·) = βˆ…)
233, 22eqtrid 2784 . . . . 5 (Β¬ 𝐷 ∈ V β†’ π‘Œ = βˆ…)
2421, 23orim12i 907 . . . 4 ((Β¬ 𝐢 ∈ V ∨ Β¬ 𝐷 ∈ V) β†’ (𝑋 = βˆ… ∨ π‘Œ = βˆ…))
25 ianor 980 . . . 4 (Β¬ (𝐢 ∈ V ∧ 𝐷 ∈ V) ↔ (Β¬ 𝐢 ∈ V ∨ Β¬ 𝐷 ∈ V))
26 xpeq0 6156 . . . 4 ((𝑋 Γ— π‘Œ) = βˆ… ↔ (𝑋 = βˆ… ∨ π‘Œ = βˆ…))
2724, 25, 263imtr4i 291 . . 3 (Β¬ (𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ (𝑋 Γ— π‘Œ) = βˆ…)
28 fnxpc 18124 . . . . . . 7 Γ—c Fn (V Γ— V)
29 fndm 6649 . . . . . . 7 ( Γ—c Fn (V Γ— V) β†’ dom Γ—c = (V Γ— V))
3028, 29ax-mp 5 . . . . . 6 dom Γ—c = (V Γ— V)
3130ndmov 7587 . . . . 5 (Β¬ (𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ (𝐢 Γ—c 𝐷) = βˆ…)
321, 31eqtrid 2784 . . . 4 (Β¬ (𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ 𝑇 = βˆ…)
3332fveq2d 6892 . . 3 (Β¬ (𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ (Baseβ€˜π‘‡) = (Baseβ€˜βˆ…))
3419, 27, 333eqtr4a 2798 . 2 (Β¬ (𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ (𝑋 Γ— π‘Œ) = (Baseβ€˜π‘‡))
3518, 34pm2.61i 182 1 (𝑋 Γ— π‘Œ) = (Baseβ€˜π‘‡)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   ∧ wa 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106  Vcvv 3474  βˆ…c0 4321  βŸ¨cop 4633   Γ— cxp 5673  dom cdm 5675   Fn wfn 6535  β€˜cfv 6540  (class class class)co 7405   ∈ cmpo 7407  1st c1st 7969  2nd c2nd 7970  Basecbs 17140  Hom chom 17204  compcco 17205   Γ—c cxpc 18116
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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-z 12555  df-dec 12674  df-slot 17111  df-ndx 17123  df-base 17141  df-hom 17217  df-cco 17218  df-xpc 18120
This theorem is referenced by:  xpchomfval  18127  xpccofval  18130  xpchom2  18134  xpcco2  18135  xpccatid  18136  1stfval  18139  2ndfval  18142  1stfcl  18145  2ndfcl  18146  prfcl  18151  prf1st  18152  prf2nd  18153  1st2ndprf  18154  catcxpccl  18155  catcxpcclOLD  18156  xpcpropd  18157  evlfcl  18171  curf1cl  18177  curf2cl  18180  curfcl  18181  uncf1  18185  uncf2  18186  uncfcurf  18188  diag11  18192  diag12  18193  diag2  18194  curf2ndf  18196  hofcl  18208  yonedalem21  18222  yonedalem22  18227  yonedalem3b  18228  yonedalem3  18229  yonedainv  18230  yonffthlem  18231
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