dfinito4 - Mathbox for Zhi Wang

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Theorem dfinito4 49199

Description: An alternate definition of df-inito 18001 using universal property. See also the "Equivalent formulations" section of https://en.wikipedia.org/wiki/Initial_and_terminal_objects 18001. (Contributed by Zhi Wang, 23-Oct-2025.)

**Assertion**
Ref	Expression
dfinito4	⊢ InitO = (𝑐 ∈ Cat ↦ ⦋(SetCat‘1_o) / 𝑑⦌⦋((1^st ‘(𝑑Δ_func𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐UP𝑑)∅))

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 18004	. . 3 ⊢ InitO Fn Cat
2		ovex 7446	. . . . . . 7 ⊢ (𝑓(𝑐UP𝑑)∅) ∈ V
3	2	dmex 7913	. . . . . 6 ⊢ dom (𝑓(𝑐UP𝑑)∅) ∈ V
4	3	csbex 5291	. . . . 5 ⊢ ⦋((1^st ‘(𝑑Δ_func𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐UP𝑑)∅) ∈ V
5	4	csbex 5291	. . . 4 ⊢ ⦋(SetCat‘1_o) / 𝑑⦌⦋((1^st ‘(𝑑Δ_func𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐UP𝑑)∅) ∈ V
6		eqid 2734	. . . 4 ⊢ (𝑐 ∈ Cat ↦ ⦋(SetCat‘1_o) / 𝑑⦌⦋((1^st ‘(𝑑Δ_func𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐UP𝑑)∅)) = (𝑐 ∈ Cat ↦ ⦋(SetCat‘1_o) / 𝑑⦌⦋((1^st ‘(𝑑Δ_func𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐UP𝑑)∅))
7	5, 6	fnmpti 6691	. . 3 ⊢ (𝑐 ∈ Cat ↦ ⦋(SetCat‘1_o) / 𝑑⦌⦋((1^st ‘(𝑑Δ_func𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐UP𝑑)∅)) Fn Cat
8		eqfnfv 7031	. . 3 ⊢ ((InitO Fn Cat ∧ (𝑐 ∈ Cat ↦ ⦋(SetCat‘1_o) / 𝑑⦌⦋((1^st ‘(𝑑Δ_func𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐UP𝑑)∅)) Fn Cat) → (InitO = (𝑐 ∈ Cat ↦ ⦋(SetCat‘1_o) / 𝑑⦌⦋((1^st ‘(𝑑Δ_func𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐UP𝑑)∅)) ↔ ∀𝑒 ∈ Cat (InitO‘𝑒) = ((𝑐 ∈ Cat ↦ ⦋(SetCat‘1_o) / 𝑑⦌⦋((1^st ‘(𝑑Δ_func𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐UP𝑑)∅))‘𝑒)))
9	1, 7, 8	mp2an 692	. 2 ⊢ (InitO = (𝑐 ∈ Cat ↦ ⦋(SetCat‘1_o) / 𝑑⦌⦋((1^st ‘(𝑑Δ_func𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐UP𝑑)∅)) ↔ ∀𝑒 ∈ Cat (InitO‘𝑒) = ((𝑐 ∈ Cat ↦ ⦋(SetCat‘1_o) / 𝑑⦌⦋((1^st ‘(𝑑Δ_func𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐UP𝑑)∅))‘𝑒))
10		eqid 2734	. . . . . 6 ⊢ (SetCat‘1_o) = (SetCat‘1_o)
11		eqid 2734	. . . . . 6 ⊢ ((1^st ‘((SetCat‘1_o)Δ_func𝑒))‘∅) = ((1^st ‘((SetCat‘1_o)Δ_func𝑒))‘∅)
12	10, 11	isinito3 49198	. . . . 5 ⊢ (𝑥 ∈ (InitO‘𝑒) ↔ 𝑥 ∈ dom (((1^st ‘((SetCat‘1_o)Δ_func𝑒))‘∅)(𝑒UP(SetCat‘1_o))∅))
13	12	eqriv 2731	. . . 4 ⊢ (InitO‘𝑒) = dom (((1^st ‘((SetCat‘1_o)Δ_func𝑒))‘∅)(𝑒UP(SetCat‘1_o))∅)
14		fvex 6899	. . . . 5 ⊢ (SetCat‘1_o) ∈ V
15		fvexd 6901	. . . . . 6 ⊢ (𝑑 = (SetCat‘1_o) → ((1^st ‘(𝑑Δ_func𝑒))‘∅) ∈ V)
16		simpl 482	. . . . . . . . 9 ⊢ ((𝑑 = (SetCat‘1_o) ∧ 𝑓 = ((1^st ‘(𝑑Δ_func𝑒))‘∅)) → 𝑑 = (SetCat‘1_o))
17	16	oveq2d 7429	. . . . . . . 8 ⊢ ((𝑑 = (SetCat‘1_o) ∧ 𝑓 = ((1^st ‘(𝑑Δ_func𝑒))‘∅)) → (𝑒UP𝑑) = (𝑒UP(SetCat‘1_o)))
18		simpr 484	. . . . . . . . 9 ⊢ ((𝑑 = (SetCat‘1_o) ∧ 𝑓 = ((1^st ‘(𝑑Δ_func𝑒))‘∅)) → 𝑓 = ((1^st ‘(𝑑Δ_func𝑒))‘∅))
19	16	fvoveq1d 7435	. . . . . . . . . 10 ⊢ ((𝑑 = (SetCat‘1_o) ∧ 𝑓 = ((1^st ‘(𝑑Δ_func𝑒))‘∅)) → (1^st ‘(𝑑Δ_func𝑒)) = (1^st ‘((SetCat‘1_o)Δ_func𝑒)))
20	19	fveq1d 6888	. . . . . . . . 9 ⊢ ((𝑑 = (SetCat‘1_o) ∧ 𝑓 = ((1^st ‘(𝑑Δ_func𝑒))‘∅)) → ((1^st ‘(𝑑Δ_func𝑒))‘∅) = ((1^st ‘((SetCat‘1_o)Δ_func𝑒))‘∅))
21	18, 20	eqtrd 2769	. . . . . . . 8 ⊢ ((𝑑 = (SetCat‘1_o) ∧ 𝑓 = ((1^st ‘(𝑑Δ_func𝑒))‘∅)) → 𝑓 = ((1^st ‘((SetCat‘1_o)Δ_func𝑒))‘∅))
22		eqidd 2735	. . . . . . . 8 ⊢ ((𝑑 = (SetCat‘1_o) ∧ 𝑓 = ((1^st ‘(𝑑Δ_func𝑒))‘∅)) → ∅ = ∅)
23	17, 21, 22	oveq123d 7434	. . . . . . 7 ⊢ ((𝑑 = (SetCat‘1_o) ∧ 𝑓 = ((1^st ‘(𝑑Δ_func𝑒))‘∅)) → (𝑓(𝑒UP𝑑)∅) = (((1^st ‘((SetCat‘1_o)Δ_func𝑒))‘∅)(𝑒UP(SetCat‘1_o))∅))
24	23	dmeqd 5896	. . . . . 6 ⊢ ((𝑑 = (SetCat‘1_o) ∧ 𝑓 = ((1^st ‘(𝑑Δ_func𝑒))‘∅)) → dom (𝑓(𝑒UP𝑑)∅) = dom (((1^st ‘((SetCat‘1_o)Δ_func𝑒))‘∅)(𝑒UP(SetCat‘1_o))∅))
25	15, 24	csbied 3915	. . . . 5 ⊢ (𝑑 = (SetCat‘1_o) → ⦋((1^st ‘(𝑑Δ_func𝑒))‘∅) / 𝑓⦌dom (𝑓(𝑒UP𝑑)∅) = dom (((1^st ‘((SetCat‘1_o)Δ_func𝑒))‘∅)(𝑒UP(SetCat‘1_o))∅))
26	14, 25	csbie 3914	. . . 4 ⊢ ⦋(SetCat‘1_o) / 𝑑⦌⦋((1^st ‘(𝑑Δ_func𝑒))‘∅) / 𝑓⦌dom (𝑓(𝑒UP𝑑)∅) = dom (((1^st ‘((SetCat‘1_o)Δ_func𝑒))‘∅)(𝑒UP(SetCat‘1_o))∅)
27	13, 26	eqtr4i 2760	. . 3 ⊢ (InitO‘𝑒) = ⦋(SetCat‘1_o) / 𝑑⦌⦋((1^st ‘(𝑑Δ_func𝑒))‘∅) / 𝑓⦌dom (𝑓(𝑒UP𝑑)∅)
28		oveq2 7421	. . . . . . . 8 ⊢ (𝑐 = 𝑒 → (𝑑Δ_func𝑐) = (𝑑Δ_func𝑒))
29	28	fveq2d 6890	. . . . . . 7 ⊢ (𝑐 = 𝑒 → (1^st ‘(𝑑Δ_func𝑐)) = (1^st ‘(𝑑Δ_func𝑒)))
30	29	fveq1d 6888	. . . . . 6 ⊢ (𝑐 = 𝑒 → ((1^st ‘(𝑑Δ_func𝑐))‘∅) = ((1^st ‘(𝑑Δ_func𝑒))‘∅))
31		oveq1 7420	. . . . . . . 8 ⊢ (𝑐 = 𝑒 → (𝑐UP𝑑) = (𝑒UP𝑑))
32	31	oveqd 7430	. . . . . . 7 ⊢ (𝑐 = 𝑒 → (𝑓(𝑐UP𝑑)∅) = (𝑓(𝑒UP𝑑)∅))
33	32	dmeqd 5896	. . . . . 6 ⊢ (𝑐 = 𝑒 → dom (𝑓(𝑐UP𝑑)∅) = dom (𝑓(𝑒UP𝑑)∅))
34	30, 33	csbeq12dv 3888	. . . . 5 ⊢ (𝑐 = 𝑒 → ⦋((1^st ‘(𝑑Δ_func𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐UP𝑑)∅) = ⦋((1^st ‘(𝑑Δ_func𝑒))‘∅) / 𝑓⦌dom (𝑓(𝑒UP𝑑)∅))
35	34	csbeq2dv 3886	. . . 4 ⊢ (𝑐 = 𝑒 → ⦋(SetCat‘1_o) / 𝑑⦌⦋((1^st ‘(𝑑Δ_func𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐UP𝑑)∅) = ⦋(SetCat‘1_o) / 𝑑⦌⦋((1^st ‘(𝑑Δ_func𝑒))‘∅) / 𝑓⦌dom (𝑓(𝑒UP𝑑)∅))
36		ovex 7446	. . . . . . 7 ⊢ (𝑓(𝑒UP𝑑)∅) ∈ V
37	36	dmex 7913	. . . . . 6 ⊢ dom (𝑓(𝑒UP𝑑)∅) ∈ V
38	37	csbex 5291	. . . . 5 ⊢ ⦋((1^st ‘(𝑑Δ_func𝑒))‘∅) / 𝑓⦌dom (𝑓(𝑒UP𝑑)∅) ∈ V
39	38	csbex 5291	. . . 4 ⊢ ⦋(SetCat‘1_o) / 𝑑⦌⦋((1^st ‘(𝑑Δ_func𝑒))‘∅) / 𝑓⦌dom (𝑓(𝑒UP𝑑)∅) ∈ V
40	35, 6, 39	fvmpt 6996	. . 3 ⊢ (𝑒 ∈ Cat → ((𝑐 ∈ Cat ↦ ⦋(SetCat‘1_o) / 𝑑⦌⦋((1^st ‘(𝑑Δ_func𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐UP𝑑)∅))‘𝑒) = ⦋(SetCat‘1_o) / 𝑑⦌⦋((1^st ‘(𝑑Δ_func𝑒))‘∅) / 𝑓⦌dom (𝑓(𝑒UP𝑑)∅))
41	27, 40	eqtr4id 2788	. 2 ⊢ (𝑒 ∈ Cat → (InitO‘𝑒) = ((𝑐 ∈ Cat ↦ ⦋(SetCat‘1_o) / 𝑑⦌⦋((1^st ‘(𝑑Δ_func𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐UP𝑑)∅))‘𝑒))
42	9, 41	mprgbir 3057	1 ⊢ InitO = (𝑐 ∈ Cat ↦ ⦋(SetCat‘1_o) / 𝑑⦌⦋((1^st ‘(𝑑Δ_func𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐UP𝑑)∅))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3050 Vcvv 3463 ⦋csb 3879 ∅c0 4313 ↦ cmpt 5205 dom cdm 5665 Fn wfn 6536 ‘cfv 6541 (class class class)co 7413 1^st c1st 7994 1_oc1o 8481 Catccat 17679 InitOcinito 17998 SetCatcsetc 18092 Δ_funccdiag 18228 UPcup 48957

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5259 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737 ax-cnex 11193 ax-resscn 11194 ax-1cn 11195 ax-icn 11196 ax-addcl 11197 ax-addrcl 11198 ax-mulcl 11199 ax-mulrcl 11200 ax-mulcom 11201 ax-addass 11202 ax-mulass 11203 ax-distr 11204 ax-i2m1 11205 ax-1ne0 11206 ax-1rid 11207 ax-rnegex 11208 ax-rrecex 11209 ax-cnre 11210 ax-pre-lttri 11211 ax-pre-lttrn 11212 ax-pre-ltadd 11213 ax-pre-mulgt0 11214

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-ot 4615 df-uni 4888 df-iun 4973 df-br 5124 df-opab 5186 df-mpt 5206 df-tr 5240 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-we 5619 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6301 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7870 df-1st 7996 df-2nd 7997 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-er 8727 df-map 8850 df-ixp 8920 df-en 8968 df-dom 8969 df-sdom 8970 df-fin 8971 df-pnf 11279 df-mnf 11280 df-xr 11281 df-ltxr 11282 df-le 11283 df-sub 11476 df-neg 11477 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12510 df-z 12597 df-dec 12717 df-uz 12861 df-fz 13530 df-struct 17167 df-slot 17202 df-ndx 17214 df-base 17231 df-hom 17298 df-cco 17299 df-cat 17683 df-cid 17684 df-func 17875 df-nat 17963 df-fuc 17964 df-inito 18001 df-setc 18093 df-xpc 18188 df-1stf 18189 df-curf 18230 df-diag 18232 df-up 48958 df-thinc 49119 df-termc 49172

This theorem is referenced by: dftermo4 49200

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