Theorem dfinito4 50967

Description: An alternate definition of df-inito 18209 using universal property. See also the "Equivalent formulations" section of https://en.wikipedia.org/wiki/Initial_and_terminal_objects 18209. (Contributed by Zhi Wang, 23-Oct-2025.)

Assertion

Ref	Expression
dfinito4	⊢ InitO = (𝑐 ∈ Cat ↦ ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅))

Distinct variable group:   𝑐,𝑑,𝑓

Proof of Theorem dfinito4

Dummy variables 𝑒 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		initofn 18212	. . 3 ⊢ InitO Fn Cat
2		ovex 7484	. . . . . . 7 ⊢ (𝑓(𝑐 UP 𝑑)∅) ∈ V
3	2	dmex 7955	. . . . . 6 ⊢ dom (𝑓(𝑐 UP 𝑑)∅) ∈ V
4	3	csbex 5271	. . . . 5 ⊢ ⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅) ∈ V
5	4	csbex 5271	. . . 4 ⊢ ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅) ∈ V
6		eqid 2763	. . . 4 ⊢ (𝑐 ∈ Cat ↦ ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅)) = (𝑐 ∈ Cat ↦ ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅))
7	5, 6	fnmpti 6712	. . 3 ⊢ (𝑐 ∈ Cat ↦ ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅)) Fn Cat
8		eqfnfv 7060	. . 3 ⊢ ((InitO Fn Cat ∧ (𝑐 ∈ Cat ↦ ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅)) Fn Cat) → (InitO = (𝑐 ∈ Cat ↦ ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅)) ↔ ∀𝑒 ∈ Cat (InitO‘𝑒) = ((𝑐 ∈ Cat ↦ ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅))‘𝑒)))
9	1, 7, 8	mp2an 705	. 2 ⊢ (InitO = (𝑐 ∈ Cat ↦ ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅)) ↔ ∀𝑒 ∈ Cat (InitO‘𝑒) = ((𝑐 ∈ Cat ↦ ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅))‘𝑒))
10		eqid 2763	. . . . . 6 ⊢ (SetCat‘1o) = (SetCat‘1o)
11		eqid 2763	. . . . . 6 ⊢ ((1st ‘((SetCat‘1o)Δfunc𝑒))‘∅) = ((1st ‘((SetCat‘1o)Δfunc𝑒))‘∅)
12	10, 11	isinito3 50966	. . . . 5 ⊢ (𝑥 ∈ (InitO‘𝑒) ↔ 𝑥 ∈ dom (((1st ‘((SetCat‘1o)Δfunc𝑒))‘∅)(𝑒 UP (SetCat‘1o))∅))
13	12	eqriv 2760	. . . 4 ⊢ (InitO‘𝑒) = dom (((1st ‘((SetCat‘1o)Δfunc𝑒))‘∅)(𝑒 UP (SetCat‘1o))∅)
14		fvex 6929	. . . . 5 ⊢ (SetCat‘1o) ∈ V
15		fvexd 6931	. . . . . 6 ⊢ (𝑑 = (SetCat‘1o) → ((1st ‘(𝑑Δfunc𝑒))‘∅) ∈ V)
16		simpl 488	. . . . . . . . 9 ⊢ ((𝑑 = (SetCat‘1o) ∧ 𝑓 = ((1st ‘(𝑑Δfunc𝑒))‘∅)) → 𝑑 = (SetCat‘1o))
17	16	oveq2d 7467	. . . . . . . 8 ⊢ ((𝑑 = (SetCat‘1o) ∧ 𝑓 = ((1st ‘(𝑑Δfunc𝑒))‘∅)) → (𝑒 UP 𝑑) = (𝑒 UP (SetCat‘1o)))
18		simpr 490	. . . . . . . . 9 ⊢ ((𝑑 = (SetCat‘1o) ∧ 𝑓 = ((1st ‘(𝑑Δfunc𝑒))‘∅)) → 𝑓 = ((1st ‘(𝑑Δfunc𝑒))‘∅))
19	16	fvoveq1d 7473	. . . . . . . . . 10 ⊢ ((𝑑 = (SetCat‘1o) ∧ 𝑓 = ((1st ‘(𝑑Δfunc𝑒))‘∅)) → (1st ‘(𝑑Δfunc𝑒)) = (1st ‘((SetCat‘1o)Δfunc𝑒)))
20	19	fveq1d 6918	. . . . . . . . 9 ⊢ ((𝑑 = (SetCat‘1o) ∧ 𝑓 = ((1st ‘(𝑑Δfunc𝑒))‘∅)) → ((1st ‘(𝑑Δfunc𝑒))‘∅) = ((1st ‘((SetCat‘1o)Δfunc𝑒))‘∅))
21	18, 20	eqtrd 2798	. . . . . . . 8 ⊢ ((𝑑 = (SetCat‘1o) ∧ 𝑓 = ((1st ‘(𝑑Δfunc𝑒))‘∅)) → 𝑓 = ((1st ‘((SetCat‘1o)Δfunc𝑒))‘∅))
22		eqidd 2764	. . . . . . . 8 ⊢ ((𝑑 = (SetCat‘1o) ∧ 𝑓 = ((1st ‘(𝑑Δfunc𝑒))‘∅)) → ∅ = ∅)
23	17, 21, 22	oveq123d 7472	. . . . . . 7 ⊢ ((𝑑 = (SetCat‘1o) ∧ 𝑓 = ((1st ‘(𝑑Δfunc𝑒))‘∅)) → (𝑓(𝑒 UP 𝑑)∅) = (((1st ‘((SetCat‘1o)Δfunc𝑒))‘∅)(𝑒 UP (SetCat‘1o))∅))
24	23	dmeqd 5899	. . . . . 6 ⊢ ((𝑑 = (SetCat‘1o) ∧ 𝑓 = ((1st ‘(𝑑Δfunc𝑒))‘∅)) → dom (𝑓(𝑒 UP 𝑑)∅) = dom (((1st ‘((SetCat‘1o)Δfunc𝑒))‘∅)(𝑒 UP (SetCat‘1o))∅))
25	15, 24	csbied 3887	. . . . 5 ⊢ (𝑑 = (SetCat‘1o) → ⦋((1st ‘(𝑑Δfunc𝑒))‘∅) / 𝑓⦌dom (𝑓(𝑒 UP 𝑑)∅) = dom (((1st ‘((SetCat‘1o)Δfunc𝑒))‘∅)(𝑒 UP (SetCat‘1o))∅))
26	14, 25	csbie 3886	. . . 4 ⊢ ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑒))‘∅) / 𝑓⦌dom (𝑓(𝑒 UP 𝑑)∅) = dom (((1st ‘((SetCat‘1o)Δfunc𝑒))‘∅)(𝑒 UP (SetCat‘1o))∅)
27	13, 26	eqtr4i 2789	. . 3 ⊢ (InitO‘𝑒) = ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑒))‘∅) / 𝑓⦌dom (𝑓(𝑒 UP 𝑑)∅)
28		oveq2 7459	. . . . . . . 8 ⊢ (𝑐 = 𝑒 → (𝑑Δfunc𝑐) = (𝑑Δfunc𝑒))
29	28	fveq2d 6920	. . . . . . 7 ⊢ (𝑐 = 𝑒 → (1st ‘(𝑑Δfunc𝑐)) = (1st ‘(𝑑Δfunc𝑒)))
30	29	fveq1d 6918	. . . . . 6 ⊢ (𝑐 = 𝑒 → ((1st ‘(𝑑Δfunc𝑐))‘∅) = ((1st ‘(𝑑Δfunc𝑒))‘∅))
31		oveq1 7458	. . . . . . . 8 ⊢ (𝑐 = 𝑒 → (𝑐 UP 𝑑) = (𝑒 UP 𝑑))
32	31	oveqd 7468	. . . . . . 7 ⊢ (𝑐 = 𝑒 → (𝑓(𝑐 UP 𝑑)∅) = (𝑓(𝑒 UP 𝑑)∅))
33	32	dmeqd 5899	. . . . . 6 ⊢ (𝑐 = 𝑒 → dom (𝑓(𝑐 UP 𝑑)∅) = dom (𝑓(𝑒 UP 𝑑)∅))
34	30, 33	csbeq12dv 3860	. . . . 5 ⊢ (𝑐 = 𝑒 → ⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅) = ⦋((1st ‘(𝑑Δfunc𝑒))‘∅) / 𝑓⦌dom (𝑓(𝑒 UP 𝑑)∅))
35	34	csbeq2dv 3858	. . . 4 ⊢ (𝑐 = 𝑒 → ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅) = ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑒))‘∅) / 𝑓⦌dom (𝑓(𝑒 UP 𝑑)∅))
36		ovex 7484	. . . . . . 7 ⊢ (𝑓(𝑒 UP 𝑑)∅) ∈ V
37	36	dmex 7955	. . . . . 6 ⊢ dom (𝑓(𝑒 UP 𝑑)∅) ∈ V
38	37	csbex 5271	. . . . 5 ⊢ ⦋((1st ‘(𝑑Δfunc𝑒))‘∅) / 𝑓⦌dom (𝑓(𝑒 UP 𝑑)∅) ∈ V
39	38	csbex 5271	. . . 4 ⊢ ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑒))‘∅) / 𝑓⦌dom (𝑓(𝑒 UP 𝑑)∅) ∈ V
40	35, 6, 39	fvmpt 7024	. . 3 ⊢ (𝑒 ∈ Cat → ((𝑐 ∈ Cat ↦ ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅))‘𝑒) = ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑒))‘∅) / 𝑓⦌dom (𝑓(𝑒 UP 𝑑)∅))
41	27, 40	eqtr4id 2817	. 2 ⊢ (𝑒 ∈ Cat → (InitO‘𝑒) = ((𝑐 ∈ Cat ↦ ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅))‘𝑒))
42	9, 41	mprgbir 3087	1 ⊢ InitO = (𝑐 ∈ Cat ↦ ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  Vcvv 3454  ⦋csb 3851  ∅c0 4281   ↦ cmpt 5192  dom cdm 5658   Fn wfn 6561  ‘cfv 6566  (class class class)co 7451  1^st c1st 8033  1_oc1o 8507  Catccat 17888  InitOcinito 18206  SetCatcsetc 18300  Δ_funccdiag 18436   UP cup 50620

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-rmo 3369  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-ixp 8964  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-cat 17892  df-cid 17893  df-func 18083  df-nat 18171  df-fuc 18172  df-inito 18209  df-setc 18301  df-xpc 18396  df-1stf 18397  df-curf 18438  df-diag 18440  df-up 50621  df-thinc 50884  df-termc 50939

This theorem is used by:  dftermo4  50968