Theorem initoo 17638

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 2738	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2738	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		id 22	. . . 4 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat)
4	1, 2, 3	isinitoi 17630	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (InitO‘𝐶)) → (𝑂 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑂(Hom ‘𝐶)𝑏)))
5	4	simpld 494	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (InitO‘𝐶)) → 𝑂 ∈ (Base‘𝐶))
6	5	ex 412	1 ⊢ (𝐶 ∈ Cat → (𝑂 ∈ (InitO‘𝐶) → 𝑂 ∈ (Base‘𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  ∃!weu 2568  ∀wral 3063  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  Hom chom 16899  Catccat 17290  InitOcinito 17612

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258  df-inito 17615

This theorem is referenced by:  iszeroi  17640  2initoinv  17641  initoeu1w  17643