Theorem dfinito4 49463

Description: An alternate definition of df-inito 17922 using universal property. See also the "Equivalent formulations" section of https://en.wikipedia.org/wiki/Initial_and_terminal_objects 17922. (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 17925	. . 3 ⊢ InitO Fn Cat
2		ovex 7402	. . . . . . 7 ⊢ (𝑓(𝑐 UP 𝑑)∅) ∈ V
3	2	dmex 7865	. . . . . 6 ⊢ dom (𝑓(𝑐 UP 𝑑)∅) ∈ V
4	3	csbex 5261	. . . . 5 ⊢ ⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅) ∈ V
5	4	csbex 5261	. . . 4 ⊢ ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅) ∈ V
6		eqid 2729	. . . 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 6643	. . 3 ⊢ (𝑐 ∈ Cat ↦ ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅)) Fn Cat
8		eqfnfv 6985	. . 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 2729	. . . . . 6 ⊢ (SetCat‘1o) = (SetCat‘1o)
11		eqid 2729	. . . . . 6 ⊢ ((1st ‘((SetCat‘1o)Δfunc𝑒))‘∅) = ((1st ‘((SetCat‘1o)Δfunc𝑒))‘∅)
12	10, 11	isinito3 49462	. . . . 5 ⊢ (𝑥 ∈ (InitO‘𝑒) ↔ 𝑥 ∈ dom (((1st ‘((SetCat‘1o)Δfunc𝑒))‘∅)(𝑒 UP (SetCat‘1o))∅))
13	12	eqriv 2726	. . . 4 ⊢ (InitO‘𝑒) = dom (((1st ‘((SetCat‘1o)Δfunc𝑒))‘∅)(𝑒 UP (SetCat‘1o))∅)
14		fvex 6853	. . . . 5 ⊢ (SetCat‘1o) ∈ V
15		fvexd 6855	. . . . . 6 ⊢ (𝑑 = (SetCat‘1o) → ((1st ‘(𝑑Δfunc𝑒))‘∅) ∈ V)
16		simpl 482	. . . . . . . . 9 ⊢ ((𝑑 = (SetCat‘1o) ∧ 𝑓 = ((1st ‘(𝑑Δfunc𝑒))‘∅)) → 𝑑 = (SetCat‘1o))
17	16	oveq2d 7385	. . . . . . . 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 7391	. . . . . . . . . 10 ⊢ ((𝑑 = (SetCat‘1o) ∧ 𝑓 = ((1st ‘(𝑑Δfunc𝑒))‘∅)) → (1st ‘(𝑑Δfunc𝑒)) = (1st ‘((SetCat‘1o)Δfunc𝑒)))
20	19	fveq1d 6842	. . . . . . . . 9 ⊢ ((𝑑 = (SetCat‘1o) ∧ 𝑓 = ((1st ‘(𝑑Δfunc𝑒))‘∅)) → ((1st ‘(𝑑Δfunc𝑒))‘∅) = ((1st ‘((SetCat‘1o)Δfunc𝑒))‘∅))
21	18, 20	eqtrd 2764	. . . . . . . 8 ⊢ ((𝑑 = (SetCat‘1o) ∧ 𝑓 = ((1st ‘(𝑑Δfunc𝑒))‘∅)) → 𝑓 = ((1st ‘((SetCat‘1o)Δfunc𝑒))‘∅))
22		eqidd 2730	. . . . . . . 8 ⊢ ((𝑑 = (SetCat‘1o) ∧ 𝑓 = ((1st ‘(𝑑Δfunc𝑒))‘∅)) → ∅ = ∅)
23	17, 21, 22	oveq123d 7390	. . . . . . 7 ⊢ ((𝑑 = (SetCat‘1o) ∧ 𝑓 = ((1st ‘(𝑑Δfunc𝑒))‘∅)) → (𝑓(𝑒 UP 𝑑)∅) = (((1st ‘((SetCat‘1o)Δfunc𝑒))‘∅)(𝑒 UP (SetCat‘1o))∅))
24	23	dmeqd 5859	. . . . . 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 3895	. . . . 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 3894	. . . 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 2755	. . 3 ⊢ (InitO‘𝑒) = ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑒))‘∅) / 𝑓⦌dom (𝑓(𝑒 UP 𝑑)∅)
28		oveq2 7377	. . . . . . . 8 ⊢ (𝑐 = 𝑒 → (𝑑Δfunc𝑐) = (𝑑Δfunc𝑒))
29	28	fveq2d 6844	. . . . . . 7 ⊢ (𝑐 = 𝑒 → (1st ‘(𝑑Δfunc𝑐)) = (1st ‘(𝑑Δfunc𝑒)))
30	29	fveq1d 6842	. . . . . 6 ⊢ (𝑐 = 𝑒 → ((1st ‘(𝑑Δfunc𝑐))‘∅) = ((1st ‘(𝑑Δfunc𝑒))‘∅))
31		oveq1 7376	. . . . . . . 8 ⊢ (𝑐 = 𝑒 → (𝑐 UP 𝑑) = (𝑒 UP 𝑑))
32	31	oveqd 7386	. . . . . . 7 ⊢ (𝑐 = 𝑒 → (𝑓(𝑐 UP 𝑑)∅) = (𝑓(𝑒 UP 𝑑)∅))
33	32	dmeqd 5859	. . . . . 6 ⊢ (𝑐 = 𝑒 → dom (𝑓(𝑐 UP 𝑑)∅) = dom (𝑓(𝑒 UP 𝑑)∅))
34	30, 33	csbeq12dv 3868	. . . . 5 ⊢ (𝑐 = 𝑒 → ⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅) = ⦋((1st ‘(𝑑Δfunc𝑒))‘∅) / 𝑓⦌dom (𝑓(𝑒 UP 𝑑)∅))
35	34	csbeq2dv 3866	. . . 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 7402	. . . . . . 7 ⊢ (𝑓(𝑒 UP 𝑑)∅) ∈ V
37	36	dmex 7865	. . . . . 6 ⊢ dom (𝑓(𝑒 UP 𝑑)∅) ∈ V
38	37	csbex 5261	. . . . 5 ⊢ ⦋((1st ‘(𝑑Δfunc𝑒))‘∅) / 𝑓⦌dom (𝑓(𝑒 UP 𝑑)∅) ∈ V
39	38	csbex 5261	. . . 4 ⊢ ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑒))‘∅) / 𝑓⦌dom (𝑓(𝑒 UP 𝑑)∅) ∈ V
40	35, 6, 39	fvmpt 6950	. . 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 2783	. 2 ⊢ (𝑒 ∈ Cat → (InitO‘𝑒) = ((𝑐 ∈ Cat ↦ ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅))‘𝑒))
42	9, 41	mprgbir 3051	1 ⊢ InitO = (𝑐 ∈ Cat ↦ ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3444  ⦋csb 3859  ∅c0 4292   ↦ cmpt 5183  dom cdm 5631   Fn wfn 6494  ‘cfv 6499  (class class class)co 7369  1^st c1st 7945  1_oc1o 8404  Catccat 17601  InitOcinito 17919  SetCatcsetc 18013  Δ_funccdiag 18149   UP cup 49135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-er 8648  df-map 8778  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-z 12506  df-dec 12626  df-uz 12770  df-fz 13445  df-struct 17093  df-slot 17128  df-ndx 17140  df-base 17156  df-hom 17220  df-cco 17221  df-cat 17605  df-cid 17606  df-func 17796  df-nat 17884  df-fuc 17885  df-inito 17922  df-setc 18014  df-xpc 18109  df-1stf 18110  df-curf 18151  df-diag 18153  df-up 49136  df-thinc 49380  df-termc 49435

This theorem is referenced by:  dftermo4  49464