Theorem initoo 18232

Description: An initial object is an object. (Contributed by AV, 14-Apr-2020.)

Assertion

Ref	Expression
initoo	⊢ (𝐶 ∈ Cat → (𝑂 ∈ (InitO‘𝐶) → 𝑂 ∈ (Base‘𝐶)))

Proof of Theorem initoo

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

Step	Hyp	Ref	Expression
1		eqid 2763	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2763	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		id 23	. . . 4 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat)
4	1, 2, 3	isinitoi 18224	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (InitO‘𝐶)) → (𝑂 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑂(Hom ‘𝐶)𝑏)))
5	4	simpld 500	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (InitO‘𝐶)) → 𝑂 ∈ (Base‘𝐶))
6	5	ex 418	1 ⊢ (𝐶 ∈ Cat → (𝑂 ∈ (InitO‘𝐶) → 𝑂 ∈ (Base‘𝐶)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2147  ∃!weu 2596  ∀wral 3080  ‘cfv 6566  (class class class)co 7451  Basecbs 17431  Hom chom 17483  Catccat 17888  InitOcinito 18206

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-iota 6520  df-fun 6568  df-fv 6574  df-ov 7454  df-inito 18209

This theorem is used by:  iszeroi  18234  2initoinv  18235  initoeu1w  18237