Theorem dfinito4 49533

Description: An alternate definition of df-inito 17886 using universal property. See also the "Equivalent formulations" section of https://en.wikipedia.org/wiki/Initial_and_terminal_objects 17886. (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 17889	. . 3 ⊢ InitO Fn Cat
2		ovex 7374	. . . . . . 7 ⊢ (𝑓(𝑐 UP 𝑑)∅) ∈ V
3	2	dmex 7834	. . . . . 6 ⊢ dom (𝑓(𝑐 UP 𝑑)∅) ∈ V
4	3	csbex 5244	. . . . 5 ⊢ ⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅) ∈ V
5	4	csbex 5244	. . . 4 ⊢ ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅) ∈ V
6		eqid 2731	. . . 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 6619	. . 3 ⊢ (𝑐 ∈ Cat ↦ ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅)) Fn Cat
8		eqfnfv 6959	. . 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 692	. 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 2731	. . . . . 6 ⊢ (SetCat‘1o) = (SetCat‘1o)
11		eqid 2731	. . . . . 6 ⊢ ((1st ‘((SetCat‘1o)Δfunc𝑒))‘∅) = ((1st ‘((SetCat‘1o)Δfunc𝑒))‘∅)
12	10, 11	isinito3 49532	. . . . 5 ⊢ (𝑥 ∈ (InitO‘𝑒) ↔ 𝑥 ∈ dom (((1st ‘((SetCat‘1o)Δfunc𝑒))‘∅)(𝑒 UP (SetCat‘1o))∅))
13	12	eqriv 2728	. . . 4 ⊢ (InitO‘𝑒) = dom (((1st ‘((SetCat‘1o)Δfunc𝑒))‘∅)(𝑒 UP (SetCat‘1o))∅)
14		fvex 6830	. . . . 5 ⊢ (SetCat‘1o) ∈ V
15		fvexd 6832	. . . . . 6 ⊢ (𝑑 = (SetCat‘1o) → ((1st ‘(𝑑Δfunc𝑒))‘∅) ∈ V)
16		simpl 482	. . . . . . . . 9 ⊢ ((𝑑 = (SetCat‘1o) ∧ 𝑓 = ((1st ‘(𝑑Δfunc𝑒))‘∅)) → 𝑑 = (SetCat‘1o))
17	16	oveq2d 7357	. . . . . . . 8 ⊢ ((𝑑 = (SetCat‘1o) ∧ 𝑓 = ((1st ‘(𝑑Δfunc𝑒))‘∅)) → (𝑒 UP 𝑑) = (𝑒 UP (SetCat‘1o)))
18		simpr 484	. . . . . . . . 9 ⊢ ((𝑑 = (SetCat‘1o) ∧ 𝑓 = ((1st ‘(𝑑Δfunc𝑒))‘∅)) → 𝑓 = ((1st ‘(𝑑Δfunc𝑒))‘∅))
19	16	fvoveq1d 7363	. . . . . . . . . 10 ⊢ ((𝑑 = (SetCat‘1o) ∧ 𝑓 = ((1st ‘(𝑑Δfunc𝑒))‘∅)) → (1st ‘(𝑑Δfunc𝑒)) = (1st ‘((SetCat‘1o)Δfunc𝑒)))
20	19	fveq1d 6819	. . . . . . . . 9 ⊢ ((𝑑 = (SetCat‘1o) ∧ 𝑓 = ((1st ‘(𝑑Δfunc𝑒))‘∅)) → ((1st ‘(𝑑Δfunc𝑒))‘∅) = ((1st ‘((SetCat‘1o)Δfunc𝑒))‘∅))
21	18, 20	eqtrd 2766	. . . . . . . 8 ⊢ ((𝑑 = (SetCat‘1o) ∧ 𝑓 = ((1st ‘(𝑑Δfunc𝑒))‘∅)) → 𝑓 = ((1st ‘((SetCat‘1o)Δfunc𝑒))‘∅))
22		eqidd 2732	. . . . . . . 8 ⊢ ((𝑑 = (SetCat‘1o) ∧ 𝑓 = ((1st ‘(𝑑Δfunc𝑒))‘∅)) → ∅ = ∅)
23	17, 21, 22	oveq123d 7362	. . . . . . 7 ⊢ ((𝑑 = (SetCat‘1o) ∧ 𝑓 = ((1st ‘(𝑑Δfunc𝑒))‘∅)) → (𝑓(𝑒 UP 𝑑)∅) = (((1st ‘((SetCat‘1o)Δfunc𝑒))‘∅)(𝑒 UP (SetCat‘1o))∅))
24	23	dmeqd 5840	. . . . . 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 3881	. . . . 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 3880	. . . 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 2757	. . 3 ⊢ (InitO‘𝑒) = ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑒))‘∅) / 𝑓⦌dom (𝑓(𝑒 UP 𝑑)∅)
28		oveq2 7349	. . . . . . . 8 ⊢ (𝑐 = 𝑒 → (𝑑Δfunc𝑐) = (𝑑Δfunc𝑒))
29	28	fveq2d 6821	. . . . . . 7 ⊢ (𝑐 = 𝑒 → (1st ‘(𝑑Δfunc𝑐)) = (1st ‘(𝑑Δfunc𝑒)))
30	29	fveq1d 6819	. . . . . 6 ⊢ (𝑐 = 𝑒 → ((1st ‘(𝑑Δfunc𝑐))‘∅) = ((1st ‘(𝑑Δfunc𝑒))‘∅))
31		oveq1 7348	. . . . . . . 8 ⊢ (𝑐 = 𝑒 → (𝑐 UP 𝑑) = (𝑒 UP 𝑑))
32	31	oveqd 7358	. . . . . . 7 ⊢ (𝑐 = 𝑒 → (𝑓(𝑐 UP 𝑑)∅) = (𝑓(𝑒 UP 𝑑)∅))
33	32	dmeqd 5840	. . . . . 6 ⊢ (𝑐 = 𝑒 → dom (𝑓(𝑐 UP 𝑑)∅) = dom (𝑓(𝑒 UP 𝑑)∅))
34	30, 33	csbeq12dv 3854	. . . . 5 ⊢ (𝑐 = 𝑒 → ⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅) = ⦋((1st ‘(𝑑Δfunc𝑒))‘∅) / 𝑓⦌dom (𝑓(𝑒 UP 𝑑)∅))
35	34	csbeq2dv 3852	. . . 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 7374	. . . . . . 7 ⊢ (𝑓(𝑒 UP 𝑑)∅) ∈ V
37	36	dmex 7834	. . . . . 6 ⊢ dom (𝑓(𝑒 UP 𝑑)∅) ∈ V
38	37	csbex 5244	. . . . 5 ⊢ ⦋((1st ‘(𝑑Δfunc𝑒))‘∅) / 𝑓⦌dom (𝑓(𝑒 UP 𝑑)∅) ∈ V
39	38	csbex 5244	. . . 4 ⊢ ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑒))‘∅) / 𝑓⦌dom (𝑓(𝑒 UP 𝑑)∅) ∈ V
40	35, 6, 39	fvmpt 6924	. . 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 2785	. 2 ⊢ (𝑒 ∈ Cat → (InitO‘𝑒) = ((𝑐 ∈ Cat ↦ ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅))‘𝑒))
42	9, 41	mprgbir 3054	1 ⊢ InitO = (𝑐 ∈ Cat ↦ ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  Vcvv 3436  ⦋csb 3845  ∅c0 4278   ↦ cmpt 5167  dom cdm 5611   Fn wfn 6471  ‘cfv 6476  (class class class)co 7341  1^st c1st 7914  1_oc1o 8373  Catccat 17565  InitOcinito 17883  SetCatcsetc 17977  Δ_funccdiag 18113   UP cup 49205

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663  ax-cnex 11057  ax-resscn 11058  ax-1cn 11059  ax-icn 11060  ax-addcl 11061  ax-addrcl 11062  ax-mulcl 11063  ax-mulrcl 11064  ax-mulcom 11065  ax-addass 11066  ax-mulass 11067  ax-distr 11068  ax-i2m1 11069  ax-1ne0 11070  ax-1rid 11071  ax-rnegex 11072  ax-rrecex 11073  ax-cnre 11074  ax-pre-lttri 11075  ax-pre-lttrn 11076  ax-pre-ltadd 11077  ax-pre-mulgt0 11078

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-tp 4576  df-op 4578  df-ot 4580  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-er 8617  df-map 8747  df-ixp 8817  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-pnf 11143  df-mnf 11144  df-xr 11145  df-ltxr 11146  df-le 11147  df-sub 11341  df-neg 11342  df-nn 12121  df-2 12183  df-3 12184  df-4 12185  df-5 12186  df-6 12187  df-7 12188  df-8 12189  df-9 12190  df-n0 12377  df-z 12464  df-dec 12584  df-uz 12728  df-fz 13403  df-struct 17053  df-slot 17088  df-ndx 17100  df-base 17116  df-hom 17180  df-cco 17181  df-cat 17569  df-cid 17570  df-func 17760  df-nat 17848  df-fuc 17849  df-inito 17886  df-setc 17978  df-xpc 18073  df-1stf 18074  df-curf 18115  df-diag 18117  df-up 49206  df-thinc 49450  df-termc 49505

This theorem is referenced by:  dftermo4  49534