Theorem setcco 18048

Description: Composition in the category of sets. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypotheses

Ref	Expression
setcbas.c	⊢ 𝐶 = (SetCat‘𝑈)
setcbas.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
setcco.o	⊢ · = (comp‘𝐶)
setcco.x	⊢ (𝜑 → 𝑋 ∈ 𝑈)
setcco.y	⊢ (𝜑 → 𝑌 ∈ 𝑈)
setcco.z	⊢ (𝜑 → 𝑍 ∈ 𝑈)
setcco.f	⊢ (𝜑 → 𝐹:𝑋⟶𝑌)
setcco.g	⊢ (𝜑 → 𝐺:𝑌⟶𝑍)

Assertion

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

Proof of Theorem setcco

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

Step	Hyp	Ref	Expression
1		setcbas.c	. . . 4 ⊢ 𝐶 = (SetCat‘𝑈)
2		setcbas.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
3		setcco.o	. . . 4 ⊢ · = (comp‘𝐶)
4	1, 2, 3	setccofval 18047	. . 3 ⊢ (𝜑 → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))
5		simprr 778	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍)
6		simprl 776	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑣 = ⟨𝑋, 𝑌⟩)
7	6	fveq2d 6838	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘𝑣) = (2nd ‘⟨𝑋, 𝑌⟩))
8		setcco.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
9		setcco.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑈)
10		op2ndg 7951	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
11	8, 9, 10	syl2anc 590	. . . . . . 7 ⊢ (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
12	11	adantr 481	. . . . . 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	5, 13	oveq12d 7381	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑧 ↑m (2nd ‘𝑣)) = (𝑍 ↑m 𝑌))
15	6	fveq2d 6838	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = (1st ‘⟨𝑋, 𝑌⟩))
16		op1stg 7950	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
17	8, 9, 16	syl2anc 590	. . . . . . 7 ⊢ (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
18	17	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
19	15, 18	eqtrd 2775	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = 𝑋)
20	13, 19	oveq12d 7381	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((2nd ‘𝑣) ↑m (1st ‘𝑣)) = (𝑌 ↑m 𝑋))
21		eqidd 2741	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓))
22	14, 20, 21	mpoeq123dv 7438	. . 3 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ (𝑍 ↑m 𝑌), 𝑓 ∈ (𝑌 ↑m 𝑋) ↦ (𝑔 ∘ 𝑓)))
23	8, 9	opelxpd 5664	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑈))
24		setcco.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
25		ovex 7396	. . . . 5 ⊢ (𝑍 ↑m 𝑌) ∈ V
26		ovex 7396	. . . . 5 ⊢ (𝑌 ↑m 𝑋) ∈ V
27	25, 26	mpoex 8028	. . . 4 ⊢ (𝑔 ∈ (𝑍 ↑m 𝑌), 𝑓 ∈ (𝑌 ↑m 𝑋) ↦ (𝑔 ∘ 𝑓)) ∈ V
28	27	a1i 11	. . 3 ⊢ (𝜑 → (𝑔 ∈ (𝑍 ↑m 𝑌), 𝑓 ∈ (𝑌 ↑m 𝑋) ↦ (𝑔 ∘ 𝑓)) ∈ V)
29	4, 22, 23, 24, 28	ovmpod 7515	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ · 𝑍) = (𝑔 ∈ (𝑍 ↑m 𝑌), 𝑓 ∈ (𝑌 ↑m 𝑋) ↦ (𝑔 ∘ 𝑓)))
30		simprl 776	. . 3 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑔 = 𝐺)
31		simprr 778	. . 3 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑓 = 𝐹)
32	30, 31	coeq12d 5813	. 2 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (𝑔 ∘ 𝑓) = (𝐺 ∘ 𝐹))
33		setcco.g	. . 3 ⊢ (𝜑 → 𝐺:𝑌⟶𝑍)
34	24, 9	elmapd 8784	. . 3 ⊢ (𝜑 → (𝐺 ∈ (𝑍 ↑m 𝑌) ↔ 𝐺:𝑌⟶𝑍))
35	33, 34	mpbird 258	. 2 ⊢ (𝜑 → 𝐺 ∈ (𝑍 ↑m 𝑌))
36		setcco.f	. . 3 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌)
37	9, 8	elmapd 8784	. . 3 ⊢ (𝜑 → (𝐹 ∈ (𝑌 ↑m 𝑋) ↔ 𝐹:𝑋⟶𝑌))
38	36, 37	mpbird 258	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑌 ↑m 𝑋))
39		coexg 7876	. . 3 ⊢ ((𝐺 ∈ (𝑍 ↑m 𝑌) ∧ 𝐹 ∈ (𝑌 ↑m 𝑋)) → (𝐺 ∘ 𝐹) ∈ V)
40	35, 38, 39	syl2anc 590	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V)
41	29, 32, 35, 38, 40	ovmpod 7515	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺 ∘ 𝐹))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3432  ⟨cop 4568   × cxp 5623   ∘ ccom 5629  ⟶wf 6488  ‘cfv 6492  (class class class)co 7363   ∈ cmpo 7365  1^st c1st 7936  2^nd c2nd 7937   ↑_m cmap 8770  compcco 17230  SetCatcsetc 18040

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-er 8640  df-map 8772  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-2 12242  df-3 12243  df-4 12244  df-5 12245  df-6 12246  df-7 12247  df-8 12248  df-9 12249  df-n0 12436  df-z 12523  df-dec 12643  df-uz 12787  df-fz 13460  df-struct 17115  df-slot 17150  df-ndx 17162  df-base 17178  df-hom 17242  df-cco 17243  df-setc 18041

This theorem is referenced by:  setccatid  18049  setcmon  18052  setcepi  18053  setcsect  18054  resssetc  18057  funcestrcsetclem9  18112  funcsetcestrclem9  18127  hofcllem  18222  yonedalem4c  18241  yonedalem3b  18243  yonedainv  18245  funcringcsetcALTV2lem9  48796  funcringcsetclem9ALTV  48819