Theorem setcco 16954

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 16953	. . 3 ⊢ (𝜑 → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ (𝑧 ↑𝑚 (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑𝑚 (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))
5		simprr 813	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍)
6		simprl 811	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑣 = ⟨𝑋, 𝑌⟩)
7	6	fveq2d 6357	. . . . . 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 7347	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
11	8, 9, 10	syl2anc 696	. . . . . . 7 ⊢ (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
12	11	adantr 472	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
13	7, 12	eqtrd 2794	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘𝑣) = 𝑌)
14	5, 13	oveq12d 6832	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑧 ↑𝑚 (2nd ‘𝑣)) = (𝑍 ↑𝑚 𝑌))
15	6	fveq2d 6357	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = (1st ‘⟨𝑋, 𝑌⟩))
16		op1stg 7346	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
17	8, 9, 16	syl2anc 696	. . . . . . 7 ⊢ (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
18	17	adantr 472	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
19	15, 18	eqtrd 2794	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = 𝑋)
20	13, 19	oveq12d 6832	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((2nd ‘𝑣) ↑𝑚 (1st ‘𝑣)) = (𝑌 ↑𝑚 𝑋))
21		eqidd 2761	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓))
22	14, 20, 21	mpt2eq123dv 6883	. . 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		opelxpi 5305	. . . 4 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑈))
24	8, 9, 23	syl2anc 696	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑈))
25		setcco.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
26		ovex 6842	. . . . 5 ⊢ (𝑍 ↑𝑚 𝑌) ∈ V
27		ovex 6842	. . . . 5 ⊢ (𝑌 ↑𝑚 𝑋) ∈ V
28	26, 27	mpt2ex 7416	. . . 4 ⊢ (𝑔 ∈ (𝑍 ↑𝑚 𝑌), 𝑓 ∈ (𝑌 ↑𝑚 𝑋) ↦ (𝑔 ∘ 𝑓)) ∈ V
29	28	a1i 11	. . 3 ⊢ (𝜑 → (𝑔 ∈ (𝑍 ↑𝑚 𝑌), 𝑓 ∈ (𝑌 ↑𝑚 𝑋) ↦ (𝑔 ∘ 𝑓)) ∈ V)
30	4, 22, 24, 25, 29	ovmpt2d 6954	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ · 𝑍) = (𝑔 ∈ (𝑍 ↑𝑚 𝑌), 𝑓 ∈ (𝑌 ↑𝑚 𝑋) ↦ (𝑔 ∘ 𝑓)))
31		simprl 811	. . 3 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑔 = 𝐺)
32		simprr 813	. . 3 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑓 = 𝐹)
33	31, 32	coeq12d 5442	. 2 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (𝑔 ∘ 𝑓) = (𝐺 ∘ 𝐹))
34		setcco.g	. . 3 ⊢ (𝜑 → 𝐺:𝑌⟶𝑍)
35	25, 9	elmapd 8039	. . 3 ⊢ (𝜑 → (𝐺 ∈ (𝑍 ↑𝑚 𝑌) ↔ 𝐺:𝑌⟶𝑍))
36	34, 35	mpbird 247	. 2 ⊢ (𝜑 → 𝐺 ∈ (𝑍 ↑𝑚 𝑌))
37		setcco.f	. . 3 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌)
38	9, 8	elmapd 8039	. . 3 ⊢ (𝜑 → (𝐹 ∈ (𝑌 ↑𝑚 𝑋) ↔ 𝐹:𝑋⟶𝑌))
39	37, 38	mpbird 247	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑌 ↑𝑚 𝑋))
40		coexg 7283	. . 3 ⊢ ((𝐺 ∈ (𝑍 ↑𝑚 𝑌) ∧ 𝐹 ∈ (𝑌 ↑𝑚 𝑋)) → (𝐺 ∘ 𝐹) ∈ V)
41	36, 39, 40	syl2anc 696	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V)
42	30, 33, 36, 39, 41	ovmpt2d 6954	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺 ∘ 𝐹))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139  Vcvv 3340  ⟨cop 4327   × cxp 5264   ∘ ccom 5270  ⟶wf 6045  ‘cfv 6049  (class class class)co 6814   ↦ cmpt2 6816  1^st c1st 7332  2^nd c2nd 7333   ↑_𝑚 cmap 8025  compcco 16175  SetCatcsetc 16946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-map 8027  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-2 11291  df-3 11292  df-4 11293  df-5 11294  df-6 11295  df-7 11296  df-8 11297  df-9 11298  df-n0 11505  df-z 11590  df-dec 11706  df-uz 11900  df-fz 12540  df-struct 16081  df-ndx 16082  df-slot 16083  df-base 16085  df-hom 16188  df-cco 16189  df-setc 16947

This theorem is referenced by:  setccatid  16955  setcmon  16958  setcepi  16959  setcsect  16960  resssetc  16963  funcestrcsetclem9  17009  funcsetcestrclem9  17024  hofcllem  17119  yonedalem4c  17138  yonedalem3b  17140  yonedainv  17142  funcringcsetcALTV2lem9  42572  funcringcsetclem9ALTV  42595