Theorem catcco 18159

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 18158	. . 3 ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓))))
6		simprl 771	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑣 = ⟨𝑋, 𝑌⟩)
7	6	fveq2d 6911	. . . . . 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 8026	. . . . . . . 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 2775	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘𝑣) = 𝑌)
14		simprr 773	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍)
15	13, 14	oveq12d 7449	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((2nd ‘𝑣) Func 𝑧) = (𝑌 Func 𝑍))
16	6	fveq2d 6911	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ( Func ‘𝑣) = ( Func ‘⟨𝑋, 𝑌⟩))
17		df-ov 7434	. . . . 5 ⊢ (𝑋 Func 𝑌) = ( Func ‘⟨𝑋, 𝑌⟩)
18	16, 17	eqtr4di 2793	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ( Func ‘𝑣) = (𝑋 Func 𝑌))
19		eqidd 2736	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔 ∘func 𝑓) = (𝑔 ∘func 𝑓))
20	15, 18, 19	mpoeq123dv 7508	. . 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 5728	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
22		catcco.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
23		ovex 7464	. . . . 5 ⊢ (𝑌 Func 𝑍) ∈ V
24		ovex 7464	. . . . 5 ⊢ (𝑋 Func 𝑌) ∈ V
25	23, 24	mpoex 8103	. . . 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 7585	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ · 𝑍) = (𝑔 ∈ (𝑌 Func 𝑍), 𝑓 ∈ (𝑋 Func 𝑌) ↦ (𝑔 ∘func 𝑓)))
28		oveq12 7440	. . 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 7466	. 2 ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ V)
33	27, 29, 30, 31, 32	ovmpod 7585	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺 ∘func 𝐹))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  Vcvv 3478  ⟨cop 4637   × cxp 5687  ‘cfv 6563  (class class class)co 7431   ∈ cmpo 7433  2^nd c2nd 8012  Basecbs 17245  compcco 17310   Func cfunc 17905   ∘_func ccofu 17907  CatCatccatc 18152

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-slot 17216  df-ndx 17228  df-base 17246  df-hom 17322  df-cco 17323  df-catc 18153

This theorem is referenced by:  catccatid  18160  resscatc  18163  catcisolem  18164  catciso  18165