Theorem setc1ocofval 49192

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 4615	. . 3 ⊢ ⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩ = ⟨⟨⟨∅, ∅⟩, ∅⟩, {⟨∅, ∅, ∅⟩}⟩
2	1	sneqi 4617	. 2 ⊢ {⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} = {⟨⟨⟨∅, ∅⟩, ∅⟩, {⟨∅, ∅, ∅⟩}⟩}
3		opex 5449	. . 3 ⊢ ⟨∅, ∅⟩ ∈ V
4		0ex 5287	. . 3 ⊢ ∅ ∈ V
5		snex 5416	. . 3 ⊢ {⟨∅, ∅, ∅⟩} ∈ V
6		funcsetc1o.1	. . . . . . . 8 ⊢ 1 = (SetCat‘1o)
7		df1o2 8495	. . . . . . . . 9 ⊢ 1o = {∅}
8	7	fveq2i 6889	. . . . . . . 8 ⊢ (SetCat‘1o) = (SetCat‘{∅})
9	6, 8	eqtri 2757	. . . . . . 7 ⊢ 1 = (SetCat‘{∅})
10		snex 5416	. . . . . . . 8 ⊢ {∅} ∈ V
11	10	a1i 11	. . . . . . 7 ⊢ (⊤ → {∅} ∈ V)
12		eqid 2734	. . . . . . 7 ⊢ (comp‘ 1 ) = (comp‘ 1 )
13	9, 11, 12	setccofval 18099	. . . . . 6 ⊢ (⊤ → (comp‘ 1 ) = (𝑣 ∈ ({∅} × {∅}), 𝑧 ∈ {∅} ↦ (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))
14	13	mptru 1546	. . . . 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 2734	. . . . . 6 ⊢ {∅} = {∅}
17		eqid 2734	. . . . . 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 7491	. . . . 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 2757	. . . 4 ⊢ (comp‘ 1 ) = (𝑣 ∈ {⟨∅, ∅⟩}, 𝑧 ∈ {∅} ↦ (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))
20	4, 4	op2ndd 8007	. . . . . 6 ⊢ (𝑣 = ⟨∅, ∅⟩ → (2nd ‘𝑣) = ∅)
21	20	oveq2d 7429	. . . . 5 ⊢ (𝑣 = ⟨∅, ∅⟩ → (𝑧 ↑m (2nd ‘𝑣)) = (𝑧 ↑m ∅))
22	4, 4	op1std 8006	. . . . . . 7 ⊢ (𝑣 = ⟨∅, ∅⟩ → (1st ‘𝑣) = ∅)
23	20, 22	oveq12d 7431	. . . . . 6 ⊢ (𝑣 = ⟨∅, ∅⟩ → ((2nd ‘𝑣) ↑m (1st ‘𝑣)) = (∅ ↑m ∅))
24		0map0sn0 8907	. . . . . 6 ⊢ (∅ ↑m ∅) = {∅}
25	23, 24	eqtrdi 2785	. . . . 5 ⊢ (𝑣 = ⟨∅, ∅⟩ → ((2nd ‘𝑣) ↑m (1st ‘𝑣)) = {∅})
26		eqidd 2735	. . . . 5 ⊢ (𝑣 = ⟨∅, ∅⟩ → (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓))
27	21, 25, 26	mpoeq123dv 7490	. . . 4 ⊢ (𝑣 = ⟨∅, ∅⟩ → (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ (𝑧 ↑m ∅), 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)))
28		oveq1 7420	. . . . . . . 8 ⊢ (𝑧 = ∅ → (𝑧 ↑m ∅) = (∅ ↑m ∅))
29	28, 24	eqtrdi 2785	. . . . . . 7 ⊢ (𝑧 = ∅ → (𝑧 ↑m ∅) = {∅})
30		eqidd 2735	. . . . . . 7 ⊢ (𝑧 = ∅ → {∅} = {∅})
31		eqidd 2735	. . . . . . 7 ⊢ (𝑧 = ∅ → (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓))
32	29, 30, 31	mpoeq123dv 7490	. . . . . 6 ⊢ (𝑧 = ∅ → (𝑔 ∈ (𝑧 ↑m ∅), 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ {∅}, 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)))
33		eqid 2734	. . . . . . . 8 ⊢ (𝑔 ∈ {∅}, 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ {∅}, 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓))
34		coeq1 5848	. . . . . . . . 9 ⊢ (𝑔 = ∅ → (𝑔 ∘ 𝑓) = (∅ ∘ 𝑓))
35		co01 6261	. . . . . . . . 9 ⊢ (∅ ∘ 𝑓) = ∅
36	34, 35	eqtrdi 2785	. . . . . . . 8 ⊢ (𝑔 = ∅ → (𝑔 ∘ 𝑓) = ∅)
37		eqidd 2735	. . . . . . . 8 ⊢ (𝑓 = ∅ → ∅ = ∅)
38	33, 36, 37	mposn 8110	. . . . . . 7 ⊢ ((∅ ∈ V ∧ ∅ ∈ V ∧ ∅ ∈ V) → (𝑔 ∈ {∅}, 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)) = {⟨⟨∅, ∅⟩, ∅⟩})
39	4, 4, 4, 38	mp3an 1462	. . . . . 6 ⊢ (𝑔 ∈ {∅}, 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)) = {⟨⟨∅, ∅⟩, ∅⟩}
40	32, 39	eqtrdi 2785	. . . . 5 ⊢ (𝑧 = ∅ → (𝑔 ∈ (𝑧 ↑m ∅), 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)) = {⟨⟨∅, ∅⟩, ∅⟩})
41		df-ot 4615	. . . . . 6 ⊢ ⟨∅, ∅, ∅⟩ = ⟨⟨∅, ∅⟩, ∅⟩
42	41	sneqi 4617	. . . . 5 ⊢ {⟨∅, ∅, ∅⟩} = {⟨⟨∅, ∅⟩, ∅⟩}
43	40, 42	eqtr4di 2787	. . . 4 ⊢ (𝑧 = ∅ → (𝑔 ∈ (𝑧 ↑m ∅), 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)) = {⟨∅, ∅, ∅⟩})
44	19, 27, 43	mposn 8110	. . 3 ⊢ ((⟨∅, ∅⟩ ∈ V ∧ ∅ ∈ V ∧ {⟨∅, ∅, ∅⟩} ∈ V) → (comp‘ 1 ) = {⟨⟨⟨∅, ∅⟩, ∅⟩, {⟨∅, ∅, ∅⟩}⟩})
45	3, 4, 5, 44	mp3an 1462	. 2 ⊢ (comp‘ 1 ) = {⟨⟨⟨∅, ∅⟩, ∅⟩, {⟨∅, ∅, ∅⟩}⟩}
46	2, 45	eqtr4i 2760	1 ⊢ {⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} = (comp‘ 1 )

Syntax hints:   = wceq 1539  ⊤wtru 1540   ∈ wcel 2107  Vcvv 3463  ∅c0 4313  {csn 4606  ⟨cop 4612  ⟨cotp 4614   × cxp 5663   ∘ ccom 5669  ‘cfv 6541  (class class class)co 7413   ∈ cmpo 7415  1^st c1st 7994  2^nd c2nd 7995  1_oc1o 8481   ↑_m cmap 8848  compcco 17286  SetCatcsetc 18092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737  ax-cnex 11193  ax-resscn 11194  ax-1cn 11195  ax-icn 11196  ax-addcl 11197  ax-addrcl 11198  ax-mulcl 11199  ax-mulrcl 11200  ax-mulcom 11201  ax-addass 11202  ax-mulass 11203  ax-distr 11204  ax-i2m1 11205  ax-1ne0 11206  ax-1rid 11207  ax-rnegex 11208  ax-rrecex 11209  ax-cnre 11210  ax-pre-lttri 11211  ax-pre-lttrn 11212  ax-pre-ltadd 11213  ax-pre-mulgt0 11214

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-ot 4615  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6301  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7870  df-1st 7996  df-2nd 7997  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-er 8727  df-map 8850  df-en 8968  df-dom 8969  df-sdom 8970  df-fin 8971  df-pnf 11279  df-mnf 11280  df-xr 11281  df-ltxr 11282  df-le 11283  df-sub 11476  df-neg 11477  df-nn 12249  df-2 12311  df-3 12312  df-4 12313  df-5 12314  df-6 12315  df-7 12316  df-8 12317  df-9 12318  df-n0 12510  df-z 12597  df-dec 12717  df-uz 12861  df-fz 13530  df-struct 17167  df-slot 17202  df-ndx 17214  df-base 17231  df-hom 17298  df-cco 17299  df-setc 18093

This theorem is referenced by:  isinito2lem  49196