Theorem termoo 18019

Description: A terminal object is an object. (Contributed by AV, 18-Apr-2020.)

Assertion

Ref	Expression
termoo	⊢ (𝐶 ∈ Cat → (𝑂 ∈ (TermO‘𝐶) → 𝑂 ∈ (Base‘𝐶)))

Proof of Theorem termoo

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

Step	Hyp	Ref	Expression
1		eqid 2735	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2735	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		id 22	. . . 4 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat)
4	1, 2, 3	istermoi 18011	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (TermO‘𝐶)) → (𝑂 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐶)𝑂)))
5	4	simpld 494	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (TermO‘𝐶)) → 𝑂 ∈ (Base‘𝐶))
6	5	ex 412	1 ⊢ (𝐶 ∈ Cat → (𝑂 ∈ (TermO‘𝐶) → 𝑂 ∈ (Base‘𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  ∃!weu 2567  ∀wral 3051  ‘cfv 6530  (class class class)co 7403  Basecbs 17226  Hom chom 17280  Catccat 17674  TermOctermo 17993

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-iota 6483  df-fun 6532  df-fv 6538  df-ov 7406  df-termo 17996

This theorem is referenced by:  2termoinv  18028  termoeu1w  18030