Theorem setc1ocofval 50305

Description: Composition in the trivial category. (Contributed by Zhi Wang, 22-Oct-2025.)

Hypothesis

Ref	Expression
funcsetc1o.1	⊢ 1 = (SetCat‘1o)

Assertion

Ref	Expression
setc1ocofval	⊢ {⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} = (comp‘ 1 )

Proof of Theorem setc1ocofval

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

Step	Hyp	Ref	Expression
1		df-ot 4600	. . 3 ⊢ ⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩ = ⟨⟨⟨∅, ∅⟩, ∅⟩, {⟨∅, ∅, ∅⟩}⟩
2	1	sneqi 4602	. 2 ⊢ {⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} = {⟨⟨⟨∅, ∅⟩, ∅⟩, {⟨∅, ∅, ∅⟩}⟩}
3		opex 5447	. . 3 ⊢ ⟨∅, ∅⟩ ∈ V
4		0ex 5272	. . 3 ⊢ ∅ ∈ V
5		snex 5412	. . 3 ⊢ {⟨∅, ∅, ∅⟩} ∈ V
6		funcsetc1o.1	. . . . . . . 8 ⊢ 1 = (SetCat‘1o)
7		df1o2 8462	. . . . . . . . 9 ⊢ 1o = {∅}
8	7	fveq2i 6888	. . . . . . . 8 ⊢ (SetCat‘1o) = (SetCat‘{∅})
9	6, 8	eqtri 2788	. . . . . . 7 ⊢ 1 = (SetCat‘{∅})
10		snex 5412	. . . . . . . 8 ⊢ {∅} ∈ V
11	10	a1i 11	. . . . . . 7 ⊢ (⊤ → {∅} ∈ V)
12		eqid 2765	. . . . . . 7 ⊢ (comp‘ 1 ) = (comp‘ 1 )
13	9, 11, 12	setccofval 18157	. . . . . 6 ⊢ (⊤ → (comp‘ 1 ) = (𝑣 ∈ ({∅} × {∅}), 𝑧 ∈ {∅} ↦ (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))
14	13	mptru 1577	. . . . 5 ⊢ (comp‘ 1 ) = (𝑣 ∈ ({∅} × {∅}), 𝑧 ∈ {∅} ↦ (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))
15	4, 4	xpsn 7141	. . . . . 6 ⊢ ({∅} × {∅}) = {⟨∅, ∅⟩}
16		eqid 2765	. . . . . 6 ⊢ {∅} = {∅}
17		eqid 2765	. . . . . 6 ⊢ (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓))
18	15, 16, 17	mpoeq123i 7492	. . . . 5 ⊢ (𝑣 ∈ ({∅} × {∅}), 𝑧 ∈ {∅} ↦ (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓))) = (𝑣 ∈ {⟨∅, ∅⟩}, 𝑧 ∈ {∅} ↦ (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))
19	14, 18	eqtri 2788	. . . 4 ⊢ (comp‘ 1 ) = (𝑣 ∈ {⟨∅, ∅⟩}, 𝑧 ∈ {∅} ↦ (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))
20	4, 4	op2ndd 7999	. . . . . 6 ⊢ (𝑣 = ⟨∅, ∅⟩ → (2nd ‘𝑣) = ∅)
21	20	oveq2d 7432	. . . . 5 ⊢ (𝑣 = ⟨∅, ∅⟩ → (𝑧 ↑m (2nd ‘𝑣)) = (𝑧 ↑m ∅))
22	4, 4	op1std 7998	. . . . . . 7 ⊢ (𝑣 = ⟨∅, ∅⟩ → (1st ‘𝑣) = ∅)
23	20, 22	oveq12d 7434	. . . . . 6 ⊢ (𝑣 = ⟨∅, ∅⟩ → ((2nd ‘𝑣) ↑m (1st ‘𝑣)) = (∅ ↑m ∅))
24		0map0sn0 8885	. . . . . 6 ⊢ (∅ ↑m ∅) = {∅}
25	23, 24	eqtrdi 2816	. . . . 5 ⊢ (𝑣 = ⟨∅, ∅⟩ → ((2nd ‘𝑣) ↑m (1st ‘𝑣)) = {∅})
26		eqidd 2766	. . . . 5 ⊢ (𝑣 = ⟨∅, ∅⟩ → (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓))
27	21, 25, 26	mpoeq123dv 7491	. . . 4 ⊢ (𝑣 = ⟨∅, ∅⟩ → (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ (𝑧 ↑m ∅), 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)))
28		oveq1 7423	. . . . . . . 8 ⊢ (𝑧 = ∅ → (𝑧 ↑m ∅) = (∅ ↑m ∅))
29	28, 24	eqtrdi 2816	. . . . . . 7 ⊢ (𝑧 = ∅ → (𝑧 ↑m ∅) = {∅})
30		eqidd 2766	. . . . . . 7 ⊢ (𝑧 = ∅ → {∅} = {∅})
31		eqidd 2766	. . . . . . 7 ⊢ (𝑧 = ∅ → (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓))
32	29, 30, 31	mpoeq123dv 7491	. . . . . 6 ⊢ (𝑧 = ∅ → (𝑔 ∈ (𝑧 ↑m ∅), 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ {∅}, 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)))
33		eqid 2765	. . . . . . . 8 ⊢ (𝑔 ∈ {∅}, 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ {∅}, 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓))
34		coeq1 5845	. . . . . . . . 9 ⊢ (𝑔 = ∅ → (𝑔 ∘ 𝑓) = (∅ ∘ 𝑓))
35		co01 6265	. . . . . . . . 9 ⊢ (∅ ∘ 𝑓) = ∅
36	34, 35	eqtrdi 2816	. . . . . . . 8 ⊢ (𝑔 = ∅ → (𝑔 ∘ 𝑓) = ∅)
37		eqidd 2766	. . . . . . . 8 ⊢ (𝑓 = ∅ → ∅ = ∅)
38	33, 36, 37	mposn 8100	. . . . . . 7 ⊢ ((∅ ∈ V ∧ ∅ ∈ V ∧ ∅ ∈ V) → (𝑔 ∈ {∅}, 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)) = {⟨⟨∅, ∅⟩, ∅⟩})
39	4, 4, 4, 38	mp3an 1490	. . . . . 6 ⊢ (𝑔 ∈ {∅}, 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)) = {⟨⟨∅, ∅⟩, ∅⟩}
40	32, 39	eqtrdi 2816	. . . . 5 ⊢ (𝑧 = ∅ → (𝑔 ∈ (𝑧 ↑m ∅), 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)) = {⟨⟨∅, ∅⟩, ∅⟩})
41		df-ot 4600	. . . . . 6 ⊢ ⟨∅, ∅, ∅⟩ = ⟨⟨∅, ∅⟩, ∅⟩
42	41	sneqi 4602	. . . . 5 ⊢ {⟨∅, ∅, ∅⟩} = {⟨⟨∅, ∅⟩, ∅⟩}
43	40, 42	eqtr4di 2818	. . . 4 ⊢ (𝑧 = ∅ → (𝑔 ∈ (𝑧 ↑m ∅), 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)) = {⟨∅, ∅, ∅⟩})
44	19, 27, 43	mposn 8100	. . 3 ⊢ ((⟨∅, ∅⟩ ∈ V ∧ ∅ ∈ V ∧ {⟨∅, ∅, ∅⟩} ∈ V) → (comp‘ 1 ) = {⟨⟨⟨∅, ∅⟩, ∅⟩, {⟨∅, ∅, ∅⟩}⟩})
45	3, 4, 5, 44	mp3an 1490	. 2 ⊢ (comp‘ 1 ) = {⟨⟨⟨∅, ∅⟩, ∅⟩, {⟨∅, ∅, ∅⟩}⟩}
46	2, 45	eqtr4i 2791	1 ⊢ {⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} = (comp‘ 1 )

This proof depends on syntax axioms:   = wceq 1570  ⊤wtru 1571   ∈ wcel 2146  Vcvv 3457  ∅c0 4286  {csn 4591  ⟨cop 4597  ⟨cotp 4599   × cxp 5661   ∘ ccom 5667  ‘cfv 6540  (class class class)co 7416   ∈ cmpo 7418  1^st c1st 7986  2^nd c2nd 7987  1_oc1o 8448   ↑_m cmap 8826  compcco 17340  SetCatcsetc 18150

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738  ax-cnex 11167  ax-resscn 11168  ax-1cn 11169  ax-icn 11170  ax-addcl 11171  ax-addrcl 11172  ax-mulcl 11173  ax-mulrcl 11174  ax-mulcom 11175  ax-addass 11176  ax-mulass 11177  ax-distr 11178  ax-i2m1 11179  ax-1ne0 11180  ax-1rid 11181  ax-rnegex 11182  ax-rrecex 11183  ax-cnre 11184  ax-pre-lttri 11185  ax-pre-lttrn 11186  ax-pre-ltadd 11187  ax-pre-mulgt0 11188

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-ot 4600  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-er 8696  df-map 8828  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11454  df-neg 11455  df-nn 12245  df-2 12314  df-3 12315  df-4 12316  df-5 12317  df-6 12318  df-7 12319  df-8 12320  df-9 12321  df-n0 12516  df-z 12603  df-dec 12724  df-uz 12875  df-fz 13548  df-struct 17225  df-slot 17260  df-ndx 17272  df-base 17288  df-hom 17352  df-cco 17353  df-setc 18151

This theorem is used by:  isinito2lem  50309  setc1onsubc  50413