Theorem catccofval 18279

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	⊢ (𝜑 → 𝑈 ∈ 𝑉)
catcco.o	⊢ · = (comp‘𝐶)

Assertion

Ref	Expression
catccofval	⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 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.

Step	Hyp	Ref	Expression
1		catcbas.c	. . . 4 ⊢ 𝐶 = (CatCat‘𝑈)
2		catcbas.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
3		catcbas.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
4	1, 3, 2	catcbas 18276	. . . 4 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat))
5		eqid 2761	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
6	1, 3, 2, 5	catchomfval 18277	. . . 4 ⊢ (𝜑 → (Hom ‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦)))
7		eqidd 2762	. . . 4 ⊢ (𝜑 → (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓))) = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓))))
8	1, 2, 4, 6, 7	catcval 18275	. . 3 ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (Hom ‘𝐶)⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))⟩})
9	8	fveq2d 6889	. 2 ⊢ (𝜑 → (comp‘𝐶) = (comp‘{⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (Hom ‘𝐶)⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))⟩}))
10		catcco.o	. 2 ⊢ · = (comp‘𝐶)
11	3	fvexi 6899	. . . . 5 ⊢ 𝐵 ∈ V
12	11, 11	xpex 7767	. . . 4 ⊢ (𝐵 × 𝐵) ∈ V
13	12, 11	mpoex 8092	. . 3 ⊢ (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓))) ∈ V
14		catstr 18135	. . . 4 ⊢ {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (Hom ‘𝐶)⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))⟩} Struct ⟨1, ;15⟩
15		ccoid 17585	. . . 4 ⊢ comp = Slot (comp‘ndx)
16		snsstp3 4779	. . . 4 ⊢ {⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))⟩} ⊆ {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (Hom ‘𝐶)⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))⟩}
17	14, 15, 16	strfv 17381	. . 3 ⊢ ((𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓))) ∈ V → (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓))) = (comp‘{⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (Hom ‘𝐶)⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))⟩}))
18	13, 17	ax-mp 5	. 2 ⊢ (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓))) = (comp‘{⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (Hom ‘𝐶)⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))⟩})
19	9, 10, 18	3eqtr4g 2821	1 ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓))))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3451  {ctp 4588  ⟨cop 4590   × cxp 5649  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  2^nd c2nd 8000  1c1 11201  5c5 12400  ;cdc 12814  ndxcnx 17371  Basecbs 17387  Hom chom 17439  compcco 17440   Func cfunc 18029   ∘_func ccofu 18031  CatCatccatc 18273

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-dec 12815  df-uz 12966  df-fz 13640  df-struct 17325  df-slot 17360  df-ndx 17372  df-base 17388  df-hom 17452  df-cco 17453  df-catc 18274

This theorem is used by:  catcco  18280