Theorem initoo2 49203

Description: An initial object is an object in the base set. (Contributed by Zhi Wang, 23-Oct-2025.)

Hypothesis

Ref	Expression
initoo2.b	⊢ 𝐵 = (Base‘𝐶)

Assertion

Ref	Expression
initoo2	⊢ (𝑂 ∈ (InitO‘𝐶) → 𝑂 ∈ 𝐵)

Proof of Theorem initoo2

Dummy variables 𝑏 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		initoo2.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
2		eqid 2730	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		initorcl 17958	. . . 4 ⊢ (𝑂 ∈ (InitO‘𝐶) → 𝐶 ∈ Cat)
4	1, 2, 3	isinitoi 17967	. . 3 ⊢ ((𝑂 ∈ (InitO‘𝐶) ∧ 𝑂 ∈ (InitO‘𝐶)) → (𝑂 ∈ 𝐵 ∧ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑂(Hom ‘𝐶)𝑏)))
5	4	anidms 566	. 2 ⊢ (𝑂 ∈ (InitO‘𝐶) → (𝑂 ∈ 𝐵 ∧ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑂(Hom ‘𝐶)𝑏)))
6	5	simpld 494	1 ⊢ (𝑂 ∈ (InitO‘𝐶) → 𝑂 ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃!weu 2562  ∀wral 3045  ‘cfv 6513  (class class class)co 7389  Basecbs 17185  Hom chom 17237  InitOcinito 17949

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 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fv 6521  df-ov 7392  df-inito 17952

This theorem is referenced by:  oppctermo  49207  isinito2  49468