Theorem setcco 17798

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 17797	. . 3 ⊢ (𝜑 → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))
5		simprr 770	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍)
6		simprl 768	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑣 = ⟨𝑋, 𝑌⟩)
7	6	fveq2d 6778	. . . . . 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 7844	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
11	8, 9, 10	syl2anc 584	. . . . . . 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 2778	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘𝑣) = 𝑌)
14	5, 13	oveq12d 7293	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑧 ↑m (2nd ‘𝑣)) = (𝑍 ↑m 𝑌))
15	6	fveq2d 6778	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = (1st ‘⟨𝑋, 𝑌⟩))
16		op1stg 7843	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
17	8, 9, 16	syl2anc 584	. . . . . . 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 2778	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = 𝑋)
20	13, 19	oveq12d 7293	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((2nd ‘𝑣) ↑m (1st ‘𝑣)) = (𝑌 ↑m 𝑋))
21		eqidd 2739	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓))
22	14, 20, 21	mpoeq123dv 7350	. . 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 5627	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑈))
24		setcco.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
25		ovex 7308	. . . . 5 ⊢ (𝑍 ↑m 𝑌) ∈ V
26		ovex 7308	. . . . 5 ⊢ (𝑌 ↑m 𝑋) ∈ V
27	25, 26	mpoex 7920	. . . 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 7425	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ · 𝑍) = (𝑔 ∈ (𝑍 ↑m 𝑌), 𝑓 ∈ (𝑌 ↑m 𝑋) ↦ (𝑔 ∘ 𝑓)))
30		simprl 768	. . 3 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑔 = 𝐺)
31		simprr 770	. . 3 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑓 = 𝐹)
32	30, 31	coeq12d 5773	. 2 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (𝑔 ∘ 𝑓) = (𝐺 ∘ 𝐹))
33		setcco.g	. . 3 ⊢ (𝜑 → 𝐺:𝑌⟶𝑍)
34	24, 9	elmapd 8629	. . 3 ⊢ (𝜑 → (𝐺 ∈ (𝑍 ↑m 𝑌) ↔ 𝐺:𝑌⟶𝑍))
35	33, 34	mpbird 256	. 2 ⊢ (𝜑 → 𝐺 ∈ (𝑍 ↑m 𝑌))
36		setcco.f	. . 3 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌)
37	9, 8	elmapd 8629	. . 3 ⊢ (𝜑 → (𝐹 ∈ (𝑌 ↑m 𝑋) ↔ 𝐹:𝑋⟶𝑌))
38	36, 37	mpbird 256	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑌 ↑m 𝑋))
39		coexg 7776	. . 3 ⊢ ((𝐺 ∈ (𝑍 ↑m 𝑌) ∧ 𝐹 ∈ (𝑌 ↑m 𝑋)) → (𝐺 ∘ 𝐹) ∈ V)
40	35, 38, 39	syl2anc 584	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V)
41	29, 32, 35, 38, 40	ovmpod 7425	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺 ∘ 𝐹))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ⟨cop 4567   × cxp 5587   ∘ ccom 5593  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  1^st c1st 7829  2^nd c2nd 7830   ↑_m cmap 8615  compcco 16974  SetCatcsetc 17790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12438  df-uz 12583  df-fz 13240  df-struct 16848  df-slot 16883  df-ndx 16895  df-base 16913  df-hom 16986  df-cco 16987  df-setc 17791

This theorem is referenced by:  setccatid  17799  setcmon  17802  setcepi  17803  setcsect  17804  resssetc  17807  funcestrcsetclem9  17865  funcsetcestrclem9  17880  hofcllem  17976  yonedalem4c  17995  yonedalem3b  17997  yonedainv  17999  funcringcsetcALTV2lem9  45602  funcringcsetclem9ALTV  45625