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

Theorem setccofval 16713
 Description: Composition in the category of sets. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
setcbas.c 𝐶 = (SetCat‘𝑈)
setcbas.u (𝜑𝑈𝑉)
setcco.o · = (comp‘𝐶)
Assertion
Ref Expression
setccofval (𝜑· = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ (𝑧𝑚 (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑𝑚 (1st𝑣)) ↦ (𝑔𝑓))))
Distinct variable groups:   𝑓,𝑔,𝑣,𝑧,𝜑   𝑣,𝑈,𝑧
Allowed substitution hints:   𝐶(𝑧,𝑣,𝑓,𝑔)   · (𝑧,𝑣,𝑓,𝑔)   𝑈(𝑓,𝑔)   𝑉(𝑧,𝑣,𝑓,𝑔)

Proof of Theorem setccofval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 setcbas.c . . 3 𝐶 = (SetCat‘𝑈)
2 setcbas.u . . 3 (𝜑𝑈𝑉)
3 eqid 2620 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
41, 2, 3setchomfval 16710 . . 3 (𝜑 → (Hom ‘𝐶) = (𝑥𝑈, 𝑦𝑈 ↦ (𝑦𝑚 𝑥)))
5 eqidd 2621 . . 3 (𝜑 → (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ (𝑧𝑚 (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑𝑚 (1st𝑣)) ↦ (𝑔𝑓))) = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ (𝑧𝑚 (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑𝑚 (1st𝑣)) ↦ (𝑔𝑓))))
61, 2, 4, 5setcval 16708 . 2 (𝜑𝐶 = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), (Hom ‘𝐶)⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ (𝑧𝑚 (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑𝑚 (1st𝑣)) ↦ (𝑔𝑓)))⟩})
7 catstr 16598 . 2 {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), (Hom ‘𝐶)⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ (𝑧𝑚 (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑𝑚 (1st𝑣)) ↦ (𝑔𝑓)))⟩} Struct ⟨1, 15⟩
8 ccoid 16058 . 2 comp = Slot (comp‘ndx)
9 snsstp3 4340 . 2 {⟨(comp‘ndx), (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ (𝑧𝑚 (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑𝑚 (1st𝑣)) ↦ (𝑔𝑓)))⟩} ⊆ {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), (Hom ‘𝐶)⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ (𝑧𝑚 (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑𝑚 (1st𝑣)) ↦ (𝑔𝑓)))⟩}
10 xpexg 6945 . . . 4 ((𝑈𝑉𝑈𝑉) → (𝑈 × 𝑈) ∈ V)
112, 2, 10syl2anc 692 . . 3 (𝜑 → (𝑈 × 𝑈) ∈ V)
12 mpt2exga 7231 . . 3 (((𝑈 × 𝑈) ∈ V ∧ 𝑈𝑉) → (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ (𝑧𝑚 (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑𝑚 (1st𝑣)) ↦ (𝑔𝑓))) ∈ V)
1311, 2, 12syl2anc 692 . 2 (𝜑 → (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ (𝑧𝑚 (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑𝑚 (1st𝑣)) ↦ (𝑔𝑓))) ∈ V)
14 setcco.o . 2 · = (comp‘𝐶)
156, 7, 8, 9, 13, 14strfv3 15889 1 (𝜑· = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ (𝑧𝑚 (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑𝑚 (1st𝑣)) ↦ (𝑔𝑓))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1481   ∈ wcel 1988  Vcvv 3195  {ctp 4172  ⟨cop 4174   × cxp 5102   ∘ ccom 5108  ‘cfv 5876  (class class class)co 6635   ↦ cmpt2 6637  1st c1st 7151  2nd c2nd 7152   ↑𝑚 cmap 7842  1c1 9922  5c5 11058  ;cdc 11478  ndxcnx 15835  Basecbs 15838  Hom chom 15933  compcco 15934  SetCatcsetc 16706 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-2 11064  df-3 11065  df-4 11066  df-5 11067  df-6 11068  df-7 11069  df-8 11070  df-9 11071  df-n0 11278  df-z 11363  df-dec 11479  df-uz 11673  df-fz 12312  df-struct 15840  df-ndx 15841  df-slot 15842  df-base 15844  df-hom 15947  df-cco 15948  df-setc 16707 This theorem is referenced by:  setcco  16714
 Copyright terms: Public domain W3C validator