Theorem setc1ocofval 49739

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 4589	. . 3 ⊢ ⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩ = ⟨⟨⟨∅, ∅⟩, ∅⟩, {⟨∅, ∅, ∅⟩}⟩
2	1	sneqi 4591	. 2 ⊢ {⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} = {⟨⟨⟨∅, ∅⟩, ∅⟩, {⟨∅, ∅, ∅⟩}⟩}
3		opex 5412	. . 3 ⊢ ⟨∅, ∅⟩ ∈ V
4		0ex 5252	. . 3 ⊢ ∅ ∈ V
5		snex 5381	. . 3 ⊢ {⟨∅, ∅, ∅⟩} ∈ V
6		funcsetc1o.1	. . . . . . . 8 ⊢ 1 = (SetCat‘1o)
7		df1o2 8404	. . . . . . . . 9 ⊢ 1o = {∅}
8	7	fveq2i 6837	. . . . . . . 8 ⊢ (SetCat‘1o) = (SetCat‘{∅})
9	6, 8	eqtri 2759	. . . . . . 7 ⊢ 1 = (SetCat‘{∅})
10		snex 5381	. . . . . . . 8 ⊢ {∅} ∈ V
11	10	a1i 11	. . . . . . 7 ⊢ (⊤ → {∅} ∈ V)
12		eqid 2736	. . . . . . 7 ⊢ (comp‘ 1 ) = (comp‘ 1 )
13	9, 11, 12	setccofval 18006	. . . . . 6 ⊢ (⊤ → (comp‘ 1 ) = (𝑣 ∈ ({∅} × {∅}), 𝑧 ∈ {∅} ↦ (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))
14	13	mptru 1548	. . . . 5 ⊢ (comp‘ 1 ) = (𝑣 ∈ ({∅} × {∅}), 𝑧 ∈ {∅} ↦ (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))
15	4, 4	xpsn 7086	. . . . . 6 ⊢ ({∅} × {∅}) = {⟨∅, ∅⟩}
16		eqid 2736	. . . . . 6 ⊢ {∅} = {∅}
17		eqid 2736	. . . . . 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 7434	. . . . 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 2759	. . . 4 ⊢ (comp‘ 1 ) = (𝑣 ∈ {⟨∅, ∅⟩}, 𝑧 ∈ {∅} ↦ (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))
20	4, 4	op2ndd 7944	. . . . . 6 ⊢ (𝑣 = ⟨∅, ∅⟩ → (2nd ‘𝑣) = ∅)
21	20	oveq2d 7374	. . . . 5 ⊢ (𝑣 = ⟨∅, ∅⟩ → (𝑧 ↑m (2nd ‘𝑣)) = (𝑧 ↑m ∅))
22	4, 4	op1std 7943	. . . . . . 7 ⊢ (𝑣 = ⟨∅, ∅⟩ → (1st ‘𝑣) = ∅)
23	20, 22	oveq12d 7376	. . . . . 6 ⊢ (𝑣 = ⟨∅, ∅⟩ → ((2nd ‘𝑣) ↑m (1st ‘𝑣)) = (∅ ↑m ∅))
24		0map0sn0 8823	. . . . . 6 ⊢ (∅ ↑m ∅) = {∅}
25	23, 24	eqtrdi 2787	. . . . 5 ⊢ (𝑣 = ⟨∅, ∅⟩ → ((2nd ‘𝑣) ↑m (1st ‘𝑣)) = {∅})
26		eqidd 2737	. . . . 5 ⊢ (𝑣 = ⟨∅, ∅⟩ → (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓))
27	21, 25, 26	mpoeq123dv 7433	. . . 4 ⊢ (𝑣 = ⟨∅, ∅⟩ → (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ (𝑧 ↑m ∅), 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)))
28		oveq1 7365	. . . . . . . 8 ⊢ (𝑧 = ∅ → (𝑧 ↑m ∅) = (∅ ↑m ∅))
29	28, 24	eqtrdi 2787	. . . . . . 7 ⊢ (𝑧 = ∅ → (𝑧 ↑m ∅) = {∅})
30		eqidd 2737	. . . . . . 7 ⊢ (𝑧 = ∅ → {∅} = {∅})
31		eqidd 2737	. . . . . . 7 ⊢ (𝑧 = ∅ → (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓))
32	29, 30, 31	mpoeq123dv 7433	. . . . . 6 ⊢ (𝑧 = ∅ → (𝑔 ∈ (𝑧 ↑m ∅), 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ {∅}, 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)))
33		eqid 2736	. . . . . . . 8 ⊢ (𝑔 ∈ {∅}, 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ {∅}, 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓))
34		coeq1 5806	. . . . . . . . 9 ⊢ (𝑔 = ∅ → (𝑔 ∘ 𝑓) = (∅ ∘ 𝑓))
35		co01 6220	. . . . . . . . 9 ⊢ (∅ ∘ 𝑓) = ∅
36	34, 35	eqtrdi 2787	. . . . . . . 8 ⊢ (𝑔 = ∅ → (𝑔 ∘ 𝑓) = ∅)
37		eqidd 2737	. . . . . . . 8 ⊢ (𝑓 = ∅ → ∅ = ∅)
38	33, 36, 37	mposn 8045	. . . . . . 7 ⊢ ((∅ ∈ V ∧ ∅ ∈ V ∧ ∅ ∈ V) → (𝑔 ∈ {∅}, 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)) = {⟨⟨∅, ∅⟩, ∅⟩})
39	4, 4, 4, 38	mp3an 1463	. . . . . 6 ⊢ (𝑔 ∈ {∅}, 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)) = {⟨⟨∅, ∅⟩, ∅⟩}
40	32, 39	eqtrdi 2787	. . . . 5 ⊢ (𝑧 = ∅ → (𝑔 ∈ (𝑧 ↑m ∅), 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)) = {⟨⟨∅, ∅⟩, ∅⟩})
41		df-ot 4589	. . . . . 6 ⊢ ⟨∅, ∅, ∅⟩ = ⟨⟨∅, ∅⟩, ∅⟩
42	41	sneqi 4591	. . . . 5 ⊢ {⟨∅, ∅, ∅⟩} = {⟨⟨∅, ∅⟩, ∅⟩}
43	40, 42	eqtr4di 2789	. . . 4 ⊢ (𝑧 = ∅ → (𝑔 ∈ (𝑧 ↑m ∅), 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)) = {⟨∅, ∅, ∅⟩})
44	19, 27, 43	mposn 8045	. . 3 ⊢ ((⟨∅, ∅⟩ ∈ V ∧ ∅ ∈ V ∧ {⟨∅, ∅, ∅⟩} ∈ V) → (comp‘ 1 ) = {⟨⟨⟨∅, ∅⟩, ∅⟩, {⟨∅, ∅, ∅⟩}⟩})
45	3, 4, 5, 44	mp3an 1463	. 2 ⊢ (comp‘ 1 ) = {⟨⟨⟨∅, ∅⟩, ∅⟩, {⟨∅, ∅, ∅⟩}⟩}
46	2, 45	eqtr4i 2762	1 ⊢ {⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} = (comp‘ 1 )

Syntax hints:   = wceq 1541  ⊤wtru 1542   ∈ wcel 2113  Vcvv 3440  ∅c0 4285  {csn 4580  ⟨cop 4586  ⟨cotp 4588   × cxp 5622   ∘ ccom 5628  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360  1^st c1st 7931  2^nd c2nd 7932  1_oc1o 8390   ↑_m cmap 8763  compcco 17189  SetCatcsetc 17999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-ot 4589  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  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 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-z 12489  df-dec 12608  df-uz 12752  df-fz 13424  df-struct 17074  df-slot 17109  df-ndx 17121  df-base 17137  df-hom 17201  df-cco 17202  df-setc 18000

This theorem is referenced by:  isinito2lem  49743  setc1onsubc  49847