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Theorem setccofval 17965
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 2736 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
41, 2, 3setchomfval 17962 . . 3 (𝜑 → (Hom ‘𝐶) = (𝑥𝑈, 𝑦𝑈 ↦ (𝑦m 𝑥)))
5 eqidd 2737 . . 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 17960 . 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 17842 . 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 17292 . 2 comp = Slot (comp‘ndx)
9 snsstp3 4777 . 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)))⟩}
102, 2xpexd 7682 . . 3 (𝜑 → (𝑈 × 𝑈) ∈ V)
11 mpoexga 8007 . . 3 (((𝑈 × 𝑈) ∈ V ∧ 𝑈𝑉) → (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ (𝑧m (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑m (1st𝑣)) ↦ (𝑔𝑓))) ∈ V)
1210, 2, 11syl2anc 584 . 2 (𝜑 → (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ (𝑧m (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑m (1st𝑣)) ↦ (𝑔𝑓))) ∈ V)
13 setcco.o . 2 · = (comp‘𝐶)
146, 7, 8, 9, 12, 13strfv3 17074 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 1541  wcel 2106  Vcvv 3444  {ctp 4589  cop 4591   × cxp 5630  ccom 5636  cfv 6494  (class class class)co 7354  cmpo 7356  1st c1st 7916  2nd c2nd 7917  m cmap 8762  1c1 11049  5c5 12208  cdc 12615  ndxcnx 17062  Basecbs 17080  Hom chom 17141  compcco 17142  SetCatcsetc 17958
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 2707  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7669  ax-cnex 11104  ax-resscn 11105  ax-1cn 11106  ax-icn 11107  ax-addcl 11108  ax-addrcl 11109  ax-mulcl 11110  ax-mulrcl 11111  ax-mulcom 11112  ax-addass 11113  ax-mulass 11114  ax-distr 11115  ax-i2m1 11116  ax-1ne0 11117  ax-1rid 11118  ax-rnegex 11119  ax-rrecex 11120  ax-cnre 11121  ax-pre-lttri 11122  ax-pre-lttrn 11123  ax-pre-ltadd 11124  ax-pre-mulgt0 11125
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3064  df-rex 3073  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-tr 5222  df-id 5530  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5587  df-we 5589  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6252  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7310  df-ov 7357  df-oprab 7358  df-mpo 7359  df-om 7800  df-1st 7918  df-2nd 7919  df-frecs 8209  df-wrecs 8240  df-recs 8314  df-rdg 8353  df-1o 8409  df-er 8645  df-en 8881  df-dom 8882  df-sdom 8883  df-fin 8884  df-pnf 11188  df-mnf 11189  df-xr 11190  df-ltxr 11191  df-le 11192  df-sub 11384  df-neg 11385  df-nn 12151  df-2 12213  df-3 12214  df-4 12215  df-5 12216  df-6 12217  df-7 12218  df-8 12219  df-9 12220  df-n0 12411  df-z 12497  df-dec 12616  df-uz 12761  df-fz 13422  df-struct 17016  df-slot 17051  df-ndx 17063  df-base 17081  df-hom 17154  df-cco 17155  df-setc 17959
This theorem is referenced by:  setcco  17966
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