Theorem initoo2 49591

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 2737	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		initorcl 17926	. . . 4 ⊢ (𝑂 ∈ (InitO‘𝐶) → 𝐶 ∈ Cat)
4	1, 2, 3	isinitoi 17935	. . 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 1542   ∈ wcel 2114  ∃!weu 2569  ∀wral 3052  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  Hom chom 17200  InitOcinito 17917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-inito 17920

This theorem is referenced by:  oppctermo  49595  isinito2  49858