Theorem dfinito4 50427

Description: An alternate definition of df-inito 18073 using universal property. See also the "Equivalent formulations" section of https://en.wikipedia.org/wiki/Initial_and_terminal_objects 18073. (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 18076	. . 3 ⊢ InitO Fn Cat
2		ovex 7446	. . . . . . 7 ⊢ (𝑓(𝑐 UP 𝑑)∅) ∈ V
3	2	dmex 7906	. . . . . 6 ⊢ dom (𝑓(𝑐 UP 𝑑)∅) ∈ V
4	3	csbex 5268	. . . . 5 ⊢ ⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅) ∈ V
5	4	csbex 5268	. . . 4 ⊢ ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅) ∈ V
6		eqid 2760	. . . 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 6675	. . 3 ⊢ (𝑐 ∈ Cat ↦ ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅)) Fn Cat
8		eqfnfv 7022	. . 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 2760	. . . . . 6 ⊢ (SetCat‘1o) = (SetCat‘1o)
11		eqid 2760	. . . . . 6 ⊢ ((1st ‘((SetCat‘1o)Δfunc𝑒))‘∅) = ((1st ‘((SetCat‘1o)Δfunc𝑒))‘∅)
12	10, 11	isinito3 50426	. . . . 5 ⊢ (𝑥 ∈ (InitO‘𝑒) ↔ 𝑥 ∈ dom (((1st ‘((SetCat‘1o)Δfunc𝑒))‘∅)(𝑒 UP (SetCat‘1o))∅))
13	12	eqriv 2757	. . . 4 ⊢ (InitO‘𝑒) = dom (((1st ‘((SetCat‘1o)Δfunc𝑒))‘∅)(𝑒 UP (SetCat‘1o))∅)
14		fvex 6891	. . . . 5 ⊢ (SetCat‘1o) ∈ V
15		fvexd 6893	. . . . . 6 ⊢ (𝑑 = (SetCat‘1o) → ((1st ‘(𝑑Δfunc𝑒))‘∅) ∈ V)
16		simpl 488	. . . . . . . . 9 ⊢ ((𝑑 = (SetCat‘1o) ∧ 𝑓 = ((1st ‘(𝑑Δfunc𝑒))‘∅)) → 𝑑 = (SetCat‘1o))
17	16	oveq2d 7429	. . . . . . . 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 7435	. . . . . . . . . 10 ⊢ ((𝑑 = (SetCat‘1o) ∧ 𝑓 = ((1st ‘(𝑑Δfunc𝑒))‘∅)) → (1st ‘(𝑑Δfunc𝑒)) = (1st ‘((SetCat‘1o)Δfunc𝑒)))
20	19	fveq1d 6880	. . . . . . . . 9 ⊢ ((𝑑 = (SetCat‘1o) ∧ 𝑓 = ((1st ‘(𝑑Δfunc𝑒))‘∅)) → ((1st ‘(𝑑Δfunc𝑒))‘∅) = ((1st ‘((SetCat‘1o)Δfunc𝑒))‘∅))
21	18, 20	eqtrd 2795	. . . . . . . 8 ⊢ ((𝑑 = (SetCat‘1o) ∧ 𝑓 = ((1st ‘(𝑑Δfunc𝑒))‘∅)) → 𝑓 = ((1st ‘((SetCat‘1o)Δfunc𝑒))‘∅))
22		eqidd 2761	. . . . . . . 8 ⊢ ((𝑑 = (SetCat‘1o) ∧ 𝑓 = ((1st ‘(𝑑Δfunc𝑒))‘∅)) → ∅ = ∅)
23	17, 21, 22	oveq123d 7434	. . . . . . 7 ⊢ ((𝑑 = (SetCat‘1o) ∧ 𝑓 = ((1st ‘(𝑑Δfunc𝑒))‘∅)) → (𝑓(𝑒 UP 𝑑)∅) = (((1st ‘((SetCat‘1o)Δfunc𝑒))‘∅)(𝑒 UP (SetCat‘1o))∅))
24	23	dmeqd 5889	. . . . . 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 3883	. . . . 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 3882	. . . 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 2786	. . 3 ⊢ (InitO‘𝑒) = ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑒))‘∅) / 𝑓⦌dom (𝑓(𝑒 UP 𝑑)∅)
28		oveq2 7421	. . . . . . . 8 ⊢ (𝑐 = 𝑒 → (𝑑Δfunc𝑐) = (𝑑Δfunc𝑒))
29	28	fveq2d 6882	. . . . . . 7 ⊢ (𝑐 = 𝑒 → (1st ‘(𝑑Δfunc𝑐)) = (1st ‘(𝑑Δfunc𝑒)))
30	29	fveq1d 6880	. . . . . 6 ⊢ (𝑐 = 𝑒 → ((1st ‘(𝑑Δfunc𝑐))‘∅) = ((1st ‘(𝑑Δfunc𝑒))‘∅))
31		oveq1 7420	. . . . . . . 8 ⊢ (𝑐 = 𝑒 → (𝑐 UP 𝑑) = (𝑒 UP 𝑑))
32	31	oveqd 7430	. . . . . . 7 ⊢ (𝑐 = 𝑒 → (𝑓(𝑐 UP 𝑑)∅) = (𝑓(𝑒 UP 𝑑)∅))
33	32	dmeqd 5889	. . . . . 6 ⊢ (𝑐 = 𝑒 → dom (𝑓(𝑐 UP 𝑑)∅) = dom (𝑓(𝑒 UP 𝑑)∅))
34	30, 33	csbeq12dv 3856	. . . . 5 ⊢ (𝑐 = 𝑒 → ⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅) = ⦋((1st ‘(𝑑Δfunc𝑒))‘∅) / 𝑓⦌dom (𝑓(𝑒 UP 𝑑)∅))
35	34	csbeq2dv 3854	. . . 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 7446	. . . . . . 7 ⊢ (𝑓(𝑒 UP 𝑑)∅) ∈ V
37	36	dmex 7906	. . . . . 6 ⊢ dom (𝑓(𝑒 UP 𝑑)∅) ∈ V
38	37	csbex 5268	. . . . 5 ⊢ ⦋((1st ‘(𝑑Δfunc𝑒))‘∅) / 𝑓⦌dom (𝑓(𝑒 UP 𝑑)∅) ∈ V
39	38	csbex 5268	. . . 4 ⊢ ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑒))‘∅) / 𝑓⦌dom (𝑓(𝑒 UP 𝑑)∅) ∈ V
40	35, 6, 39	fvmpt 6986	. . 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 2814	. 2 ⊢ (𝑒 ∈ Cat → (InitO‘𝑒) = ((𝑐 ∈ Cat ↦ ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅))‘𝑒))
42	9, 41	mprgbir 3083	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 2145  ∀wral 3076  Vcvv 3450  ⦋csb 3847  ∅c0 4279   ↦ cmpt 5186  dom cdm 5655   Fn wfn 6528  ‘cfv 6533  (class class class)co 7413  1^st c1st 7984  1_oc1o 8448  Catccat 17752  InitOcinito 18070  SetCatcsetc 18164  Δ_funccdiag 18300   UP cup 50099

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-ot 4593  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-er 8696  df-map 8828  df-ixp 8905  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-nn 12258  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12529  df-z 12616  df-dec 12737  df-uz 12888  df-fz 13562  df-struct 17239  df-slot 17274  df-ndx 17286  df-base 17302  df-hom 17366  df-cco 17367  df-cat 17756  df-cid 17757  df-func 17947  df-nat 18035  df-fuc 18036  df-inito 18073  df-setc 18165  df-xpc 18260  df-1stf 18261  df-curf 18302  df-diag 18304  df-up 50100  df-thinc 50344  df-termc 50399

This theorem is used by:  dftermo4  50428