Theorem setc1ocofval 49456

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 4594	. . 3 ⊢ ⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩ = ⟨⟨⟨∅, ∅⟩, ∅⟩, {⟨∅, ∅, ∅⟩}⟩
2	1	sneqi 4596	. 2 ⊢ {⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} = {⟨⟨⟨∅, ∅⟩, ∅⟩, {⟨∅, ∅, ∅⟩}⟩}
3		opex 5419	. . 3 ⊢ ⟨∅, ∅⟩ ∈ V
4		0ex 5257	. . 3 ⊢ ∅ ∈ V
5		snex 5386	. . 3 ⊢ {⟨∅, ∅, ∅⟩} ∈ V
6		funcsetc1o.1	. . . . . . . 8 ⊢ 1 = (SetCat‘1o)
7		df1o2 8418	. . . . . . . . 9 ⊢ 1o = {∅}
8	7	fveq2i 6843	. . . . . . . 8 ⊢ (SetCat‘1o) = (SetCat‘{∅})
9	6, 8	eqtri 2752	. . . . . . 7 ⊢ 1 = (SetCat‘{∅})
10		snex 5386	. . . . . . . 8 ⊢ {∅} ∈ V
11	10	a1i 11	. . . . . . 7 ⊢ (⊤ → {∅} ∈ V)
12		eqid 2729	. . . . . . 7 ⊢ (comp‘ 1 ) = (comp‘ 1 )
13	9, 11, 12	setccofval 18020	. . . . . 6 ⊢ (⊤ → (comp‘ 1 ) = (𝑣 ∈ ({∅} × {∅}), 𝑧 ∈ {∅} ↦ (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))
14	13	mptru 1547	. . . . 5 ⊢ (comp‘ 1 ) = (𝑣 ∈ ({∅} × {∅}), 𝑧 ∈ {∅} ↦ (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))
15	4, 4	xpsn 7095	. . . . . 6 ⊢ ({∅} × {∅}) = {⟨∅, ∅⟩}
16		eqid 2729	. . . . . 6 ⊢ {∅} = {∅}
17		eqid 2729	. . . . . 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 7445	. . . . 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 2752	. . . 4 ⊢ (comp‘ 1 ) = (𝑣 ∈ {⟨∅, ∅⟩}, 𝑧 ∈ {∅} ↦ (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))
20	4, 4	op2ndd 7958	. . . . . 6 ⊢ (𝑣 = ⟨∅, ∅⟩ → (2nd ‘𝑣) = ∅)
21	20	oveq2d 7385	. . . . 5 ⊢ (𝑣 = ⟨∅, ∅⟩ → (𝑧 ↑m (2nd ‘𝑣)) = (𝑧 ↑m ∅))
22	4, 4	op1std 7957	. . . . . . 7 ⊢ (𝑣 = ⟨∅, ∅⟩ → (1st ‘𝑣) = ∅)
23	20, 22	oveq12d 7387	. . . . . 6 ⊢ (𝑣 = ⟨∅, ∅⟩ → ((2nd ‘𝑣) ↑m (1st ‘𝑣)) = (∅ ↑m ∅))
24		0map0sn0 8835	. . . . . 6 ⊢ (∅ ↑m ∅) = {∅}
25	23, 24	eqtrdi 2780	. . . . 5 ⊢ (𝑣 = ⟨∅, ∅⟩ → ((2nd ‘𝑣) ↑m (1st ‘𝑣)) = {∅})
26		eqidd 2730	. . . . 5 ⊢ (𝑣 = ⟨∅, ∅⟩ → (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓))
27	21, 25, 26	mpoeq123dv 7444	. . . 4 ⊢ (𝑣 = ⟨∅, ∅⟩ → (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ (𝑧 ↑m ∅), 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)))
28		oveq1 7376	. . . . . . . 8 ⊢ (𝑧 = ∅ → (𝑧 ↑m ∅) = (∅ ↑m ∅))
29	28, 24	eqtrdi 2780	. . . . . . 7 ⊢ (𝑧 = ∅ → (𝑧 ↑m ∅) = {∅})
30		eqidd 2730	. . . . . . 7 ⊢ (𝑧 = ∅ → {∅} = {∅})
31		eqidd 2730	. . . . . . 7 ⊢ (𝑧 = ∅ → (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓))
32	29, 30, 31	mpoeq123dv 7444	. . . . . 6 ⊢ (𝑧 = ∅ → (𝑔 ∈ (𝑧 ↑m ∅), 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ {∅}, 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)))
33		eqid 2729	. . . . . . . 8 ⊢ (𝑔 ∈ {∅}, 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ {∅}, 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓))
34		coeq1 5811	. . . . . . . . 9 ⊢ (𝑔 = ∅ → (𝑔 ∘ 𝑓) = (∅ ∘ 𝑓))
35		co01 6222	. . . . . . . . 9 ⊢ (∅ ∘ 𝑓) = ∅
36	34, 35	eqtrdi 2780	. . . . . . . 8 ⊢ (𝑔 = ∅ → (𝑔 ∘ 𝑓) = ∅)
37		eqidd 2730	. . . . . . . 8 ⊢ (𝑓 = ∅ → ∅ = ∅)
38	33, 36, 37	mposn 8059	. . . . . . 7 ⊢ ((∅ ∈ V ∧ ∅ ∈ V ∧ ∅ ∈ V) → (𝑔 ∈ {∅}, 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)) = {⟨⟨∅, ∅⟩, ∅⟩})
39	4, 4, 4, 38	mp3an 1463	. . . . . 6 ⊢ (𝑔 ∈ {∅}, 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)) = {⟨⟨∅, ∅⟩, ∅⟩}
40	32, 39	eqtrdi 2780	. . . . 5 ⊢ (𝑧 = ∅ → (𝑔 ∈ (𝑧 ↑m ∅), 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)) = {⟨⟨∅, ∅⟩, ∅⟩})
41		df-ot 4594	. . . . . 6 ⊢ ⟨∅, ∅, ∅⟩ = ⟨⟨∅, ∅⟩, ∅⟩
42	41	sneqi 4596	. . . . 5 ⊢ {⟨∅, ∅, ∅⟩} = {⟨⟨∅, ∅⟩, ∅⟩}
43	40, 42	eqtr4di 2782	. . . 4 ⊢ (𝑧 = ∅ → (𝑔 ∈ (𝑧 ↑m ∅), 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)) = {⟨∅, ∅, ∅⟩})
44	19, 27, 43	mposn 8059	. . 3 ⊢ ((⟨∅, ∅⟩ ∈ V ∧ ∅ ∈ V ∧ {⟨∅, ∅, ∅⟩} ∈ V) → (comp‘ 1 ) = {⟨⟨⟨∅, ∅⟩, ∅⟩, {⟨∅, ∅, ∅⟩}⟩})
45	3, 4, 5, 44	mp3an 1463	. 2 ⊢ (comp‘ 1 ) = {⟨⟨⟨∅, ∅⟩, ∅⟩, {⟨∅, ∅, ∅⟩}⟩}
46	2, 45	eqtr4i 2755	1 ⊢ {⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} = (comp‘ 1 )

Syntax hints:   = wceq 1540  ⊤wtru 1541   ∈ wcel 2109  Vcvv 3444  ∅c0 4292  {csn 4585  ⟨cop 4591  ⟨cotp 4593   × cxp 5629   ∘ ccom 5635  ‘cfv 6499  (class class class)co 7369   ∈ cmpo 7371  1^st c1st 7945  2^nd c2nd 7946  1_oc1o 8404   ↑_m cmap 8776  compcco 17208  SetCatcsetc 18013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-er 8648  df-map 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-z 12506  df-dec 12626  df-uz 12770  df-fz 13445  df-struct 17093  df-slot 17128  df-ndx 17140  df-base 17156  df-hom 17220  df-cco 17221  df-setc 18014

This theorem is referenced by:  isinito2lem  49460  setc1onsubc  49564