MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpcbas Structured version   Visualization version   GIF version

Theorem xpcbas 17424
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 2801 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
5 eqid 2801 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
6 eqid 2801 . . . 4 (comp‘𝐶) = (comp‘𝐶)
7 eqid 2801 . . . 4 (comp‘𝐷) = (comp‘𝐷)
8 simpl 486 . . . 4 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐶 ∈ V)
9 simpr 488 . . . 4 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐷 ∈ V)
10 eqidd 2802 . . . 4 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑋 × 𝑌) = (𝑋 × 𝑌))
11 eqidd 2802 . . . 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 2802 . . . 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 17423 . . 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 6663 . . . . 5 𝑋 ∈ V
153fvexi 6663 . . . . 5 𝑌 ∈ V
1614, 15xpex 7460 . . . 4 (𝑋 × 𝑌) ∈ V
1716a1i 11 . . 3 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑋 × 𝑌) ∈ V)
1813, 17estrreslem1 17383 . 2 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑋 × 𝑌) = (Base‘𝑇))
19 base0 16532 . . 3 ∅ = (Base‘∅)
20 fvprc 6642 . . . . . 6 𝐶 ∈ V → (Base‘𝐶) = ∅)
212, 20syl5eq 2848 . . . . 5 𝐶 ∈ V → 𝑋 = ∅)
22 fvprc 6642 . . . . . 6 𝐷 ∈ V → (Base‘𝐷) = ∅)
233, 22syl5eq 2848 . . . . 5 𝐷 ∈ V → 𝑌 = ∅)
2421, 23orim12i 906 . . . 4 ((¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V) → (𝑋 = ∅ ∨ 𝑌 = ∅))
25 ianor 979 . . . 4 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V))
26 xpeq0 5988 . . . 4 ((𝑋 × 𝑌) = ∅ ↔ (𝑋 = ∅ ∨ 𝑌 = ∅))
2724, 25, 263imtr4i 295 . . 3 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑋 × 𝑌) = ∅)
28 fnxpc 17422 . . . . . . 7 ×c Fn (V × V)
29 fndm 6429 . . . . . . 7 ( ×c Fn (V × V) → dom ×c = (V × V))
3028, 29ax-mp 5 . . . . . 6 dom ×c = (V × V)
3130ndmov 7316 . . . . 5 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 ×c 𝐷) = ∅)
321, 31syl5eq 2848 . . . 4 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑇 = ∅)
3332fveq2d 6653 . . 3 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (Base‘𝑇) = (Base‘∅))
3419, 27, 333eqtr4a 2862 . 2 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑋 × 𝑌) = (Base‘𝑇))
3518, 34pm2.61i 185 1 (𝑋 × 𝑌) = (Base‘𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 399  wo 844   = wceq 1538  wcel 2112  Vcvv 3444  c0 4246  cop 4534   × cxp 5521  dom cdm 5523   Fn wfn 6323  cfv 6328  (class class class)co 7139  cmpo 7141  1st c1st 7673  2nd c2nd 7674  Basecbs 16479  Hom chom 16572  compcco 16573   ×c cxpc 17414
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-ndx 16482  df-slot 16483  df-base 16485  df-hom 16585  df-cco 16586  df-xpc 17418
This theorem is referenced by:  xpchomfval  17425  xpccofval  17428  xpchom2  17432  xpcco2  17433  xpccatid  17434  1stfval  17437  2ndfval  17440  1stfcl  17443  2ndfcl  17444  prfcl  17449  prf1st  17450  prf2nd  17451  1st2ndprf  17452  catcxpccl  17453  xpcpropd  17454  evlfcl  17468  curf1cl  17474  curf2cl  17477  curfcl  17478  uncf1  17482  uncf2  17483  uncfcurf  17485  diag11  17489  diag12  17490  diag2  17491  curf2ndf  17493  hofcl  17505  yonedalem21  17519  yonedalem22  17524  yonedalem3b  17525  yonedalem3  17526  yonedainv  17527  yonffthlem  17528
  Copyright terms: Public domain W3C validator