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Theorem catccofval 17135
 Description: Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
catcbas.c 𝐶 = (CatCat‘𝑈)
catcbas.b 𝐵 = (Base‘𝐶)
catcbas.u (𝜑𝑈𝑉)
catcco.o · = (comp‘𝐶)
Assertion
Ref Expression
catccofval (𝜑· = (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔func 𝑓))))
Distinct variable groups:   𝑧,𝑣,𝐵   𝑓,𝑔,𝑣,𝑧,𝜑   𝑣,𝑈,𝑧
Allowed substitution hints:   𝐵(𝑓,𝑔)   𝐶(𝑧,𝑣,𝑓,𝑔)   · (𝑧,𝑣,𝑓,𝑔)   𝑈(𝑓,𝑔)   𝑉(𝑧,𝑣,𝑓,𝑔)

Proof of Theorem catccofval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 catcbas.c . . . 4 𝐶 = (CatCat‘𝑈)
2 catcbas.u . . . 4 (𝜑𝑈𝑉)
3 catcbas.b . . . . 5 𝐵 = (Base‘𝐶)
41, 3, 2catcbas 17132 . . . 4 (𝜑𝐵 = (𝑈 ∩ Cat))
5 eqid 2778 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
61, 3, 2, 5catchomfval 17133 . . . 4 (𝜑 → (Hom ‘𝐶) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 Func 𝑦)))
7 eqidd 2779 . . . 4 (𝜑 → (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔func 𝑓))) = (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔func 𝑓))))
81, 2, 4, 6, 7catcval 17131 . . 3 (𝜑𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (Hom ‘𝐶)⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔func 𝑓)))⟩})
98fveq2d 6450 . 2 (𝜑 → (comp‘𝐶) = (comp‘{⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (Hom ‘𝐶)⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔func 𝑓)))⟩}))
10 catcco.o . 2 · = (comp‘𝐶)
113fvexi 6460 . . . . 5 𝐵 ∈ V
1211, 11xpex 7240 . . . 4 (𝐵 × 𝐵) ∈ V
1312, 11mpt2ex 7527 . . 3 (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔func 𝑓))) ∈ V
14 catstr 17002 . . . 4 {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (Hom ‘𝐶)⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔func 𝑓)))⟩} Struct ⟨1, 15⟩
15 ccoid 16463 . . . 4 comp = Slot (comp‘ndx)
16 snsstp3 4580 . . . 4 {⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔func 𝑓)))⟩} ⊆ {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (Hom ‘𝐶)⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔func 𝑓)))⟩}
1714, 15, 16strfv 16303 . . 3 ((𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔func 𝑓))) ∈ V → (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔func 𝑓))) = (comp‘{⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (Hom ‘𝐶)⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔func 𝑓)))⟩}))
1813, 17ax-mp 5 . 2 (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔func 𝑓))) = (comp‘{⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (Hom ‘𝐶)⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔func 𝑓)))⟩})
199, 10, 183eqtr4g 2839 1 (𝜑· = (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔func 𝑓))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1601   ∈ wcel 2107  Vcvv 3398  {ctp 4402  ⟨cop 4404   × cxp 5353  ‘cfv 6135  (class class class)co 6922   ↦ cmpt2 6924  2nd c2nd 7444  1c1 10273  5c5 11433  ;cdc 11845  ndxcnx 16252  Basecbs 16255  Hom chom 16349  compcco 16350   Func cfunc 16899   ∘func ccofu 16901  CatCatccatc 17129 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-oadd 7847  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-2 11438  df-3 11439  df-4 11440  df-5 11441  df-6 11442  df-7 11443  df-8 11444  df-9 11445  df-n0 11643  df-z 11729  df-dec 11846  df-uz 11993  df-fz 12644  df-struct 16257  df-ndx 16258  df-slot 16259  df-base 16261  df-hom 16362  df-cco 16363  df-catc 17130 This theorem is referenced by:  catcco  17136
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