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Theorem xpcbas 18071
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 2733 . . . 4 (Hom β€˜πΆ) = (Hom β€˜πΆ)
5 eqid 2733 . . . 4 (Hom β€˜π·) = (Hom β€˜π·)
6 eqid 2733 . . . 4 (compβ€˜πΆ) = (compβ€˜πΆ)
7 eqid 2733 . . . 4 (compβ€˜π·) = (compβ€˜π·)
8 simpl 484 . . . 4 ((𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ 𝐢 ∈ V)
9 simpr 486 . . . 4 ((𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ 𝐷 ∈ V)
10 eqidd 2734 . . . 4 ((𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ (𝑋 Γ— π‘Œ) = (𝑋 Γ— π‘Œ))
11 eqidd 2734 . . . 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 2734 . . . 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 18070 . . 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 6857 . . . . 5 𝑋 ∈ V
153fvexi 6857 . . . . 5 π‘Œ ∈ V
1614, 15xpex 7688 . . . 4 (𝑋 Γ— π‘Œ) ∈ V
1716a1i 11 . . 3 ((𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ (𝑋 Γ— π‘Œ) ∈ V)
1813, 17estrreslem1 18029 . 2 ((𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ (𝑋 Γ— π‘Œ) = (Baseβ€˜π‘‡))
19 base0 17093 . . 3 βˆ… = (Baseβ€˜βˆ…)
20 fvprc 6835 . . . . . 6 (Β¬ 𝐢 ∈ V β†’ (Baseβ€˜πΆ) = βˆ…)
212, 20eqtrid 2785 . . . . 5 (Β¬ 𝐢 ∈ V β†’ 𝑋 = βˆ…)
22 fvprc 6835 . . . . . 6 (Β¬ 𝐷 ∈ V β†’ (Baseβ€˜π·) = βˆ…)
233, 22eqtrid 2785 . . . . 5 (Β¬ 𝐷 ∈ V β†’ π‘Œ = βˆ…)
2421, 23orim12i 908 . . . 4 ((Β¬ 𝐢 ∈ V ∨ Β¬ 𝐷 ∈ V) β†’ (𝑋 = βˆ… ∨ π‘Œ = βˆ…))
25 ianor 981 . . . 4 (Β¬ (𝐢 ∈ V ∧ 𝐷 ∈ V) ↔ (Β¬ 𝐢 ∈ V ∨ Β¬ 𝐷 ∈ V))
26 xpeq0 6113 . . . 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 18069 . . . . . . 7 Γ—c Fn (V Γ— V)
29 fndm 6606 . . . . . . 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 7539 . . . . 5 (Β¬ (𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ (𝐢 Γ—c 𝐷) = βˆ…)
321, 31eqtrid 2785 . . . 4 (Β¬ (𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ 𝑇 = βˆ…)
3332fveq2d 6847 . . 3 (Β¬ (𝐢 ∈ V ∧ 𝐷 ∈ V) β†’ (Baseβ€˜π‘‡) = (Baseβ€˜βˆ…))
3419, 27, 333eqtr4a 2799 . 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 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107  Vcvv 3444  βˆ…c0 4283  βŸ¨cop 4593   Γ— cxp 5632  dom cdm 5634   Fn wfn 6492  β€˜cfv 6497  (class class class)co 7358   ∈ cmpo 7360  1st c1st 7920  2nd c2nd 7921  Basecbs 17088  Hom chom 17149  compcco 17150   Γ—c cxpc 18061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-3 12222  df-4 12223  df-5 12224  df-6 12225  df-7 12226  df-8 12227  df-9 12228  df-n0 12419  df-z 12505  df-dec 12624  df-slot 17059  df-ndx 17071  df-base 17089  df-hom 17162  df-cco 17163  df-xpc 18065
This theorem is referenced by:  xpchomfval  18072  xpccofval  18075  xpchom2  18079  xpcco2  18080  xpccatid  18081  1stfval  18084  2ndfval  18087  1stfcl  18090  2ndfcl  18091  prfcl  18096  prf1st  18097  prf2nd  18098  1st2ndprf  18099  catcxpccl  18100  catcxpcclOLD  18101  xpcpropd  18102  evlfcl  18116  curf1cl  18122  curf2cl  18125  curfcl  18126  uncf1  18130  uncf2  18131  uncfcurf  18133  diag11  18137  diag12  18138  diag2  18139  curf2ndf  18141  hofcl  18153  yonedalem21  18167  yonedalem22  18172  yonedalem3b  18173  yonedalem3  18174  yonedainv  18175  yonffthlem  18176
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