Theorem setcco 18150

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 18149	. . 3 ⊢ (𝜑 → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))
5		simprr 772	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍)
6		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑣 = ⟨𝑋, 𝑌⟩)
7	6	fveq2d 6924	. . . . . 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 8043	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
11	8, 9, 10	syl2anc 583	. . . . . . 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 2780	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘𝑣) = 𝑌)
14	5, 13	oveq12d 7466	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑧 ↑m (2nd ‘𝑣)) = (𝑍 ↑m 𝑌))
15	6	fveq2d 6924	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = (1st ‘⟨𝑋, 𝑌⟩))
16		op1stg 8042	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
17	8, 9, 16	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
18	17	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
19	15, 18	eqtrd 2780	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = 𝑋)
20	13, 19	oveq12d 7466	. . . 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 7525	. . 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 5739	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑈))
24		setcco.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
25		ovex 7481	. . . . 5 ⊢ (𝑍 ↑m 𝑌) ∈ V
26		ovex 7481	. . . . 5 ⊢ (𝑌 ↑m 𝑋) ∈ V
27	25, 26	mpoex 8120	. . . 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 7602	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ · 𝑍) = (𝑔 ∈ (𝑍 ↑m 𝑌), 𝑓 ∈ (𝑌 ↑m 𝑋) ↦ (𝑔 ∘ 𝑓)))
30		simprl 770	. . 3 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑔 = 𝐺)
31		simprr 772	. . 3 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑓 = 𝐹)
32	30, 31	coeq12d 5889	. 2 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (𝑔 ∘ 𝑓) = (𝐺 ∘ 𝐹))
33		setcco.g	. . 3 ⊢ (𝜑 → 𝐺:𝑌⟶𝑍)
34	24, 9	elmapd 8898	. . 3 ⊢ (𝜑 → (𝐺 ∈ (𝑍 ↑m 𝑌) ↔ 𝐺:𝑌⟶𝑍))
35	33, 34	mpbird 257	. 2 ⊢ (𝜑 → 𝐺 ∈ (𝑍 ↑m 𝑌))
36		setcco.f	. . 3 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌)
37	9, 8	elmapd 8898	. . 3 ⊢ (𝜑 → (𝐹 ∈ (𝑌 ↑m 𝑋) ↔ 𝐹:𝑋⟶𝑌))
38	36, 37	mpbird 257	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑌 ↑m 𝑋))
39		coexg 7969	. . 3 ⊢ ((𝐺 ∈ (𝑍 ↑m 𝑌) ∧ 𝐹 ∈ (𝑌 ↑m 𝑋)) → (𝐺 ∘ 𝐹) ∈ V)
40	35, 38, 39	syl2anc 583	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V)
41	29, 32, 35, 38, 40	ovmpod 7602	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺 ∘ 𝐹))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488  ⟨cop 4654   × cxp 5698   ∘ ccom 5704  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  1^st c1st 8028  2^nd c2nd 8029   ↑_m cmap 8884  compcco 17323  SetCatcsetc 18142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-dec 12759  df-uz 12904  df-fz 13568  df-struct 17194  df-slot 17229  df-ndx 17241  df-base 17259  df-hom 17335  df-cco 17336  df-setc 18143

This theorem is referenced by:  setccatid  18151  setcmon  18154  setcepi  18155  setcsect  18156  resssetc  18159  funcestrcsetclem9  18217  funcsetcestrclem9  18232  hofcllem  18328  yonedalem4c  18347  yonedalem3b  18349  yonedainv  18351  funcringcsetcALTV2lem9  48021  funcringcsetclem9ALTV  48044