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Theorem termoid 16857
Description: For a terminal object, the identity arrow is the one and only morphism of the object to the object itself. (Contributed by AV, 18-Apr-2020.)
Hypotheses
Ref Expression
isinitoi.b 𝐵 = (Base‘𝐶)
isinitoi.h 𝐻 = (Hom ‘𝐶)
isinitoi.c (𝜑𝐶 ∈ Cat)
Assertion
Ref Expression
termoid ((𝜑𝑂 ∈ (TermO‘𝐶)) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)})

Proof of Theorem termoid
Dummy variables 𝑜 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isinitoi.b . . 3 𝐵 = (Base‘𝐶)
2 isinitoi.h . . 3 𝐻 = (Hom ‘𝐶)
3 isinitoi.c . . 3 (𝜑𝐶 ∈ Cat)
41, 2, 3istermoi 16855 . 2 ((𝜑𝑂 ∈ (TermO‘𝐶)) → (𝑂𝐵 ∧ ∀𝑜𝐵 ∃! ∈ (𝑜𝐻𝑂)))
5 oveq1 6820 . . . . . . . 8 (𝑜 = 𝑂 → (𝑜𝐻𝑂) = (𝑂𝐻𝑂))
65eleq2d 2825 . . . . . . 7 (𝑜 = 𝑂 → ( ∈ (𝑜𝐻𝑂) ↔ ∈ (𝑂𝐻𝑂)))
76eubidv 2627 . . . . . 6 (𝑜 = 𝑂 → (∃! ∈ (𝑜𝐻𝑂) ↔ ∃! ∈ (𝑂𝐻𝑂)))
87rspcv 3445 . . . . 5 (𝑂𝐵 → (∀𝑜𝐵 ∃! ∈ (𝑜𝐻𝑂) → ∃! ∈ (𝑂𝐻𝑂)))
98adantl 473 . . . 4 (((𝜑𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂𝐵) → (∀𝑜𝐵 ∃! ∈ (𝑜𝐻𝑂) → ∃! ∈ (𝑂𝐻𝑂)))
10 eusn 4409 . . . . 5 (∃! ∈ (𝑂𝐻𝑂) ↔ ∃(𝑂𝐻𝑂) = {})
11 eqid 2760 . . . . . . . . 9 (Id‘𝐶) = (Id‘𝐶)
123ad2antrr 764 . . . . . . . . 9 (((𝜑𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂𝐵) → 𝐶 ∈ Cat)
13 simpr 479 . . . . . . . . 9 (((𝜑𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂𝐵) → 𝑂𝐵)
141, 2, 11, 12, 13catidcl 16544 . . . . . . . 8 (((𝜑𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂𝐵) → ((Id‘𝐶)‘𝑂) ∈ (𝑂𝐻𝑂))
15 fvex 6362 . . . . . . . . . . . . 13 ((Id‘𝐶)‘𝑂) ∈ V
1615elsn 4336 . . . . . . . . . . . 12 (((Id‘𝐶)‘𝑂) ∈ {} ↔ ((Id‘𝐶)‘𝑂) = )
17 eqcom 2767 . . . . . . . . . . . 12 (((Id‘𝐶)‘𝑂) = = ((Id‘𝐶)‘𝑂))
18 vex 3343 . . . . . . . . . . . . 13 ∈ V
19 sneqbg 4519 . . . . . . . . . . . . . 14 ( ∈ V → ({} = {((Id‘𝐶)‘𝑂)} ↔ = ((Id‘𝐶)‘𝑂)))
2019bicomd 213 . . . . . . . . . . . . 13 ( ∈ V → ( = ((Id‘𝐶)‘𝑂) ↔ {} = {((Id‘𝐶)‘𝑂)}))
2118, 20ax-mp 5 . . . . . . . . . . . 12 ( = ((Id‘𝐶)‘𝑂) ↔ {} = {((Id‘𝐶)‘𝑂)})
2216, 17, 213bitri 286 . . . . . . . . . . 11 (((Id‘𝐶)‘𝑂) ∈ {} ↔ {} = {((Id‘𝐶)‘𝑂)})
2322biimpi 206 . . . . . . . . . 10 (((Id‘𝐶)‘𝑂) ∈ {} → {} = {((Id‘𝐶)‘𝑂)})
2423a1i 11 . . . . . . . . 9 ((𝑂𝐻𝑂) = {} → (((Id‘𝐶)‘𝑂) ∈ {} → {} = {((Id‘𝐶)‘𝑂)}))
25 eleq2 2828 . . . . . . . . 9 ((𝑂𝐻𝑂) = {} → (((Id‘𝐶)‘𝑂) ∈ (𝑂𝐻𝑂) ↔ ((Id‘𝐶)‘𝑂) ∈ {}))
26 eqeq1 2764 . . . . . . . . 9 ((𝑂𝐻𝑂) = {} → ((𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)} ↔ {} = {((Id‘𝐶)‘𝑂)}))
2724, 25, 263imtr4d 283 . . . . . . . 8 ((𝑂𝐻𝑂) = {} → (((Id‘𝐶)‘𝑂) ∈ (𝑂𝐻𝑂) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
2814, 27syl5 34 . . . . . . 7 ((𝑂𝐻𝑂) = {} → (((𝜑𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂𝐵) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
2928exlimiv 2007 . . . . . 6 (∃(𝑂𝐻𝑂) = {} → (((𝜑𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂𝐵) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
3029com12 32 . . . . 5 (((𝜑𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂𝐵) → (∃(𝑂𝐻𝑂) = {} → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
3110, 30syl5bi 232 . . . 4 (((𝜑𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂𝐵) → (∃! ∈ (𝑂𝐻𝑂) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
329, 31syld 47 . . 3 (((𝜑𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂𝐵) → (∀𝑜𝐵 ∃! ∈ (𝑜𝐻𝑂) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
3332expimpd 630 . 2 ((𝜑𝑂 ∈ (TermO‘𝐶)) → ((𝑂𝐵 ∧ ∀𝑜𝐵 ∃! ∈ (𝑜𝐻𝑂)) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
344, 33mpd 15 1 ((𝜑𝑂 ∈ (TermO‘𝐶)) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wex 1853  wcel 2139  ∃!weu 2607  wral 3050  Vcvv 3340  {csn 4321  cfv 6049  (class class class)co 6813  Basecbs 16059  Hom chom 16154  Catccat 16526  Idccid 16527  TermOctermo 16840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-cat 16530  df-cid 16531  df-termo 16843
This theorem is referenced by:  2termoinv  16868
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