Theorem catcco 17361

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 17360	. . 3 ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓))))
6		simprl 769	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑣 = ⟨𝑋, 𝑌⟩)
7	6	fveq2d 6674	. . . . . 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 7702	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
11	8, 9, 10	syl2anc 586	. . . . . . 7 ⊢ (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
12	11	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
13	7, 12	eqtrd 2856	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘𝑣) = 𝑌)
14		simprr 771	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍)
15	13, 14	oveq12d 7174	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((2nd ‘𝑣) Func 𝑧) = (𝑌 Func 𝑍))
16	6	fveq2d 6674	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ( Func ‘𝑣) = ( Func ‘⟨𝑋, 𝑌⟩))
17		df-ov 7159	. . . . 5 ⊢ (𝑋 Func 𝑌) = ( Func ‘⟨𝑋, 𝑌⟩)
18	16, 17	syl6eqr 2874	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ( Func ‘𝑣) = (𝑋 Func 𝑌))
19		eqidd 2822	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔 ∘func 𝑓) = (𝑔 ∘func 𝑓))
20	15, 18, 19	mpoeq123dv 7229	. . 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 5593	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
22		catcco.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
23		ovex 7189	. . . . 5 ⊢ (𝑌 Func 𝑍) ∈ V
24		ovex 7189	. . . . 5 ⊢ (𝑋 Func 𝑌) ∈ V
25	23, 24	mpoex 7777	. . . 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 7302	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ · 𝑍) = (𝑔 ∈ (𝑌 Func 𝑍), 𝑓 ∈ (𝑋 Func 𝑌) ↦ (𝑔 ∘func 𝑓)))
28		oveq12 7165	. . 3 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝐹) → (𝑔 ∘func 𝑓) = (𝐺 ∘func 𝐹))
29	28	adantl 484	. 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 7191	. 2 ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ V)
33	27, 29, 30, 31, 32	ovmpod 7302	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺 ∘func 𝐹))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Vcvv 3494  ⟨cop 4573   × cxp 5553  ‘cfv 6355  (class class class)co 7156   ∈ cmpo 7158  2^nd c2nd 7688  Basecbs 16483  compcco 16577   Func cfunc 17124   ∘_func ccofu 17126  CatCatccatc 17354

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-oadd 8106  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-2 11701  df-3 11702  df-4 11703  df-5 11704  df-6 11705  df-7 11706  df-8 11707  df-9 11708  df-n0 11899  df-z 11983  df-dec 12100  df-uz 12245  df-fz 12894  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-hom 16589  df-cco 16590  df-catc 17355

This theorem is referenced by:  catccatid  17362  resscatc  17365  catcisolem  17366  catciso  17367