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Theorem xpcbas 18140
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 2726 . . . 4 (Hom β€˜πΆ) = (Hom β€˜πΆ)
5 eqid 2726 . . . 4 (Hom β€˜π·) = (Hom β€˜π·)
6 eqid 2726 . . . 4 (compβ€˜πΆ) = (compβ€˜πΆ)
7 eqid 2726 . . . 4 (compβ€˜π·) = (compβ€˜π·)
8 simpl 482 . . . 4 ((𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ 𝐢 ∈ V)
9 simpr 484 . . . 4 ((𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ 𝐷 ∈ V)
10 eqidd 2727 . . . 4 ((𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ (𝑋 Γ— π‘Œ) = (𝑋 Γ— π‘Œ))
11 eqidd 2727 . . . 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 2727 . . . 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 18139 . . 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 6898 . . . . 5 𝑋 ∈ V
153fvexi 6898 . . . . 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 18098 . 2 ((𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ (𝑋 Γ— π‘Œ) = (Baseβ€˜π‘‡))
19 base0 17156 . . 3 βˆ… = (Baseβ€˜βˆ…)
20 fvprc 6876 . . . . . 6 (Β¬ 𝐢 ∈ V β†’ (Baseβ€˜πΆ) = βˆ…)
212, 20eqtrid 2778 . . . . 5 (Β¬ 𝐢 ∈ V β†’ 𝑋 = βˆ…)
22 fvprc 6876 . . . . . 6 (Β¬ 𝐷 ∈ V β†’ (Baseβ€˜π·) = βˆ…)
233, 22eqtrid 2778 . . . . 5 (Β¬ 𝐷 ∈ V β†’ π‘Œ = βˆ…)
2421, 23orim12i 905 . . . 4 ((Β¬ 𝐢 ∈ V ∨ Β¬ 𝐷 ∈ V) β†’ (𝑋 = βˆ… ∨ π‘Œ = βˆ…))
25 ianor 978 . . . 4 (Β¬ (𝐢 ∈ V ∧ 𝐷 ∈ V) ↔ (Β¬ 𝐢 ∈ V ∨ Β¬ 𝐷 ∈ V))
26 xpeq0 6152 . . . 4 ((𝑋 Γ— π‘Œ) = βˆ… ↔ (𝑋 = βˆ… ∨ π‘Œ = βˆ…))
2724, 25, 263imtr4i 292 . . 3 (Β¬ (𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ (𝑋 Γ— π‘Œ) = βˆ…)
28 fnxpc 18138 . . . . . . 7 Γ—c Fn (V Γ— V)
29 fndm 6645 . . . . . . 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 2778 . . . 4 (Β¬ (𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ 𝑇 = βˆ…)
3332fveq2d 6888 . . 3 (Β¬ (𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ (Baseβ€˜π‘‡) = (Baseβ€˜βˆ…))
3419, 27, 333eqtr4a 2792 . 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 395   ∨ wo 844   = wceq 1533   ∈ wcel 2098  Vcvv 3468  βˆ…c0 4317  βŸ¨cop 4629   Γ— cxp 5667  dom cdm 5669   Fn wfn 6531  β€˜cfv 6536  (class class class)co 7404   ∈ cmpo 7406  1st c1st 7969  2nd c2nd 7970  Basecbs 17151  Hom chom 17215  compcco 17216   Γ—c cxpc 18130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-nn 12214  df-2 12276  df-3 12277  df-4 12278  df-5 12279  df-6 12280  df-7 12281  df-8 12282  df-9 12283  df-n0 12474  df-z 12560  df-dec 12679  df-slot 17122  df-ndx 17134  df-base 17152  df-hom 17228  df-cco 17229  df-xpc 18134
This theorem is referenced by:  xpchomfval  18141  xpccofval  18144  xpchom2  18148  xpcco2  18149  xpccatid  18150  1stfval  18153  2ndfval  18156  1stfcl  18159  2ndfcl  18160  prfcl  18165  prf1st  18166  prf2nd  18167  1st2ndprf  18168  catcxpccl  18169  catcxpcclOLD  18170  xpcpropd  18171  evlfcl  18185  curf1cl  18191  curf2cl  18194  curfcl  18195  uncf1  18199  uncf2  18200  uncfcurf  18202  diag11  18206  diag12  18207  diag2  18208  curf2ndf  18210  hofcl  18222  yonedalem21  18236  yonedalem22  18241  yonedalem3b  18242  yonedalem3  18243  yonedainv  18244  yonffthlem  18245
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