Theorem setc1ocofval 50960

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 4864	. . 3 ⊢ ⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩ = ⟨⟨⟨∅, ∅⟩, ∅⟩, {⟨∅, ∅, ∅⟩}⟩
2	1	sneqi 4595	. 2 ⊢ {⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} = {⟨⟨⟨∅, ∅⟩, ∅⟩, {⟨∅, ∅, ∅⟩}⟩}
3		opex 5439	. . 3 ⊢ ⟨∅, ∅⟩ ∈ V
4		0ex 5267	. . 3 ⊢ ∅ ∈ V
5		snex 5404	. . 3 ⊢ {⟨∅, ∅, ∅⟩} ∈ V
6		funcsetc1o.1	. . . . . . . 8 ⊢ 1 = (SetCat‘1o)
7		df1o2 8521	. . . . . . . . 9 ⊢ 1o = {∅}
8	7	fveq2i 6919	. . . . . . . 8 ⊢ (SetCat‘1o) = (SetCat‘{∅})
9	6, 8	eqtri 2786	. . . . . . 7 ⊢ 1 = (SetCat‘{∅})
10		snex 5404	. . . . . . . 8 ⊢ {∅} ∈ V
11	10	a1i 11	. . . . . . 7 ⊢ (⊤ → {∅} ∈ V)
12		eqid 2763	. . . . . . 7 ⊢ (comp‘ 1 ) = (comp‘ 1 )
13	9, 11, 12	setccofval 18307	. . . . . 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 7173	. . . . . 6 ⊢ ({∅} × {∅}) = {⟨∅, ∅⟩}
16		eqid 2763	. . . . . 6 ⊢ {∅} = {∅}
17		eqid 2763	. . . . . 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 7527	. . . . 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 2786	. . . 4 ⊢ (comp‘ 1 ) = (𝑣 ∈ {⟨∅, ∅⟩}, 𝑧 ∈ {∅} ↦ (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))
20	4, 4	op2ndd 8046	. . . . . 6 ⊢ (𝑣 = ⟨∅, ∅⟩ → (2nd ‘𝑣) = ∅)
21	20	oveq2d 7467	. . . . 5 ⊢ (𝑣 = ⟨∅, ∅⟩ → (𝑧 ↑m (2nd ‘𝑣)) = (𝑧 ↑m ∅))
22	4, 4	op1std 8045	. . . . . . 7 ⊢ (𝑣 = ⟨∅, ∅⟩ → (1st ‘𝑣) = ∅)
23	20, 22	oveq12d 7469	. . . . . 6 ⊢ (𝑣 = ⟨∅, ∅⟩ → ((2nd ‘𝑣) ↑m (1st ‘𝑣)) = (∅ ↑m ∅))
24		0map0sn0 8951	. . . . . 6 ⊢ (∅ ↑m ∅) = {∅}
25	23, 24	eqtrdi 2814	. . . . 5 ⊢ (𝑣 = ⟨∅, ∅⟩ → ((2nd ‘𝑣) ↑m (1st ‘𝑣)) = {∅})
26		eqidd 2764	. . . . 5 ⊢ (𝑣 = ⟨∅, ∅⟩ → (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓))
27	21, 25, 26	mpoeq123dv 7526	. . . 4 ⊢ (𝑣 = ⟨∅, ∅⟩ → (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ (𝑧 ↑m ∅), 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)))
28		oveq1 7458	. . . . . . . 8 ⊢ (𝑧 = ∅ → (𝑧 ↑m ∅) = (∅ ↑m ∅))
29	28, 24	eqtrdi 2814	. . . . . . 7 ⊢ (𝑧 = ∅ → (𝑧 ↑m ∅) = {∅})
30		eqidd 2764	. . . . . . 7 ⊢ (𝑧 = ∅ → {∅} = {∅})
31		eqidd 2764	. . . . . . 7 ⊢ (𝑧 = ∅ → (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓))
32	29, 30, 31	mpoeq123dv 7526	. . . . . 6 ⊢ (𝑧 = ∅ → (𝑔 ∈ (𝑧 ↑m ∅), 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ {∅}, 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)))
33		eqid 2763	. . . . . . . 8 ⊢ (𝑔 ∈ {∅}, 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ {∅}, 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓))
34		coeq1 5846	. . . . . . . . 9 ⊢ (𝑔 = ∅ → (𝑔 ∘ 𝑓) = (∅ ∘ 𝑓))
35		co01 6284	. . . . . . . . 9 ⊢ (∅ ∘ 𝑓) = ∅
36	34, 35	eqtrdi 2814	. . . . . . . 8 ⊢ (𝑔 = ∅ → (𝑔 ∘ 𝑓) = ∅)
37		eqidd 2764	. . . . . . . 8 ⊢ (𝑓 = ∅ → ∅ = ∅)
38	33, 36, 37	mposn 8151	. . . . . . 7 ⊢ ((∅ ∈ V ∧ ∅ ∈ V ∧ ∅ ∈ V) → (𝑔 ∈ {∅}, 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)) = {⟨⟨∅, ∅⟩, ∅⟩})
39	4, 4, 4, 38	mp3an 1491	. . . . . 6 ⊢ (𝑔 ∈ {∅}, 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)) = {⟨⟨∅, ∅⟩, ∅⟩}
40	32, 39	eqtrdi 2814	. . . . 5 ⊢ (𝑧 = ∅ → (𝑔 ∈ (𝑧 ↑m ∅), 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)) = {⟨⟨∅, ∅⟩, ∅⟩})
41		df-ot 4864	. . . . . 6 ⊢ ⟨∅, ∅, ∅⟩ = ⟨⟨∅, ∅⟩, ∅⟩
42	41	sneqi 4595	. . . . 5 ⊢ {⟨∅, ∅, ∅⟩} = {⟨⟨∅, ∅⟩, ∅⟩}
43	40, 42	eqtr4di 2816	. . . 4 ⊢ (𝑧 = ∅ → (𝑔 ∈ (𝑧 ↑m ∅), 𝑓 ∈ {∅} ↦ (𝑔 ∘ 𝑓)) = {⟨∅, ∅, ∅⟩})
44	19, 27, 43	mposn 8151	. . 3 ⊢ ((⟨∅, ∅⟩ ∈ V ∧ ∅ ∈ V ∧ {⟨∅, ∅, ∅⟩} ∈ V) → (comp‘ 1 ) = {⟨⟨⟨∅, ∅⟩, ∅⟩, {⟨∅, ∅, ∅⟩}⟩})
45	3, 4, 5, 44	mp3an 1491	. 2 ⊢ (comp‘ 1 ) = {⟨⟨⟨∅, ∅⟩, ∅⟩, {⟨∅, ∅, ∅⟩}⟩}
46	2, 45	eqtr4i 2789	1 ⊢ {⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} = (comp‘ 1 )

This proof depends on syntax axioms:   = wceq 1570  ⊤wtru 1571   ∈ wcel 2147  Vcvv 3454  ∅c0 4281  {csn 4588  ⟨cop 4840  ⟨cotp 4863   × cxp 5656   ∘ ccom 5662  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453  1^st c1st 8033  2^nd c2nd 8034  1_oc1o 8507   ↑_m cmap 8885  compcco 17484  SetCatcsetc 18300

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-ot 4864  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-er 8755  df-map 8887  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-2 12445  df-3 12446  df-4 12447  df-5 12448  df-6 12449  df-7 12450  df-8 12451  df-9 12452  df-n0 12647  df-z 12734  df-dec 12855  df-uz 13006  df-fz 13680  df-struct 17369  df-slot 17404  df-ndx 17416  df-base 17432  df-hom 17497  df-cco 17498  df-setc 18301

This theorem is used by:  isinito2lem  50964  setc1onsubc  51071