Theorem catcco 18012

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)
catcco.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
catcco.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
catcco.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
catcco.f	⊢ (𝜑 → 𝐹 ∈ (𝑋 Func 𝑌))
catcco.g	⊢ (𝜑 → 𝐺 ∈ (𝑌 Func 𝑍))

Assertion

Ref	Expression
catcco	⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺 ∘func 𝐹))

Proof of Theorem catcco

Dummy variables 𝑣 𝑧 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		catcbas.c	. . . 4 ⊢ 𝐶 = (CatCat‘𝑈)
2		catcbas.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
3		catcbas.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4		catcco.o	. . . 4 ⊢ · = (comp‘𝐶)
5	1, 2, 3, 4	catccofval 18011	. . 3 ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓))))
6		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑣 = ⟨𝑋, 𝑌⟩)
7	6	fveq2d 6826	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘𝑣) = (2nd ‘⟨𝑋, 𝑌⟩))
8		catcco.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9		catcco.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
10		op2ndg 7934	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
11	8, 9, 10	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
12	11	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
13	7, 12	eqtrd 2766	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘𝑣) = 𝑌)
14		simprr 772	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍)
15	13, 14	oveq12d 7364	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((2nd ‘𝑣) Func 𝑧) = (𝑌 Func 𝑍))
16	6	fveq2d 6826	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ( Func ‘𝑣) = ( Func ‘⟨𝑋, 𝑌⟩))
17		df-ov 7349	. . . . 5 ⊢ (𝑋 Func 𝑌) = ( Func ‘⟨𝑋, 𝑌⟩)
18	16, 17	eqtr4di 2784	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ( Func ‘𝑣) = (𝑋 Func 𝑌))
19		eqidd 2732	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔 ∘func 𝑓) = (𝑔 ∘func 𝑓))
20	15, 18, 19	mpoeq123dv 7421	. . 3 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)) = (𝑔 ∈ (𝑌 Func 𝑍), 𝑓 ∈ (𝑋 Func 𝑌) ↦ (𝑔 ∘func 𝑓)))
21	8, 9	opelxpd 5653	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
22		catcco.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
23		ovex 7379	. . . . 5 ⊢ (𝑌 Func 𝑍) ∈ V
24		ovex 7379	. . . . 5 ⊢ (𝑋 Func 𝑌) ∈ V
25	23, 24	mpoex 8011	. . . 4 ⊢ (𝑔 ∈ (𝑌 Func 𝑍), 𝑓 ∈ (𝑋 Func 𝑌) ↦ (𝑔 ∘func 𝑓)) ∈ V
26	25	a1i 11	. . 3 ⊢ (𝜑 → (𝑔 ∈ (𝑌 Func 𝑍), 𝑓 ∈ (𝑋 Func 𝑌) ↦ (𝑔 ∘func 𝑓)) ∈ V)
27	5, 20, 21, 22, 26	ovmpod 7498	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ · 𝑍) = (𝑔 ∈ (𝑌 Func 𝑍), 𝑓 ∈ (𝑋 Func 𝑌) ↦ (𝑔 ∘func 𝑓)))
28		oveq12 7355	. . 3 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝐹) → (𝑔 ∘func 𝑓) = (𝐺 ∘func 𝐹))
29	28	adantl 481	. 2 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (𝑔 ∘func 𝑓) = (𝐺 ∘func 𝐹))
30		catcco.g	. 2 ⊢ (𝜑 → 𝐺 ∈ (𝑌 Func 𝑍))
31		catcco.f	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋 Func 𝑌))
32		ovexd 7381	. 2 ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ V)
33	27, 29, 30, 31, 32	ovmpod 7498	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺 ∘func 𝐹))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436  ⟨cop 4579   × cxp 5612  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  2^nd c2nd 7920  Basecbs 17120  compcco 17173   Func cfunc 17761   ∘_func ccofu 17763  CatCatccatc 18005

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-2 12188  df-3 12189  df-4 12190  df-5 12191  df-6 12192  df-7 12193  df-8 12194  df-9 12195  df-n0 12382  df-z 12469  df-dec 12589  df-uz 12733  df-fz 13408  df-struct 17058  df-slot 17093  df-ndx 17105  df-base 17121  df-hom 17185  df-cco 17186  df-catc 18006

This theorem is referenced by:  catccatid  18013  resscatc  18016  catcisolem  18017  catciso  18018  catcsect  49509