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Theorem termoid 17728
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 17726 . 2 ((𝜑𝑂 ∈ (TermO‘𝐶)) → (𝑂𝐵 ∧ ∀𝑜𝐵 ∃! ∈ (𝑜𝐻𝑂)))
5 oveq1 7279 . . . . . . . 8 (𝑜 = 𝑂 → (𝑜𝐻𝑂) = (𝑂𝐻𝑂))
65eleq2d 2826 . . . . . . 7 (𝑜 = 𝑂 → ( ∈ (𝑜𝐻𝑂) ↔ ∈ (𝑂𝐻𝑂)))
76eubidv 2588 . . . . . 6 (𝑜 = 𝑂 → (∃! ∈ (𝑜𝐻𝑂) ↔ ∃! ∈ (𝑂𝐻𝑂)))
87rspcv 3556 . . . . 5 (𝑂𝐵 → (∀𝑜𝐵 ∃! ∈ (𝑜𝐻𝑂) → ∃! ∈ (𝑂𝐻𝑂)))
98adantl 482 . . . 4 (((𝜑𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂𝐵) → (∀𝑜𝐵 ∃! ∈ (𝑜𝐻𝑂) → ∃! ∈ (𝑂𝐻𝑂)))
10 eusn 4672 . . . . 5 (∃! ∈ (𝑂𝐻𝑂) ↔ ∃(𝑂𝐻𝑂) = {})
11 eqid 2740 . . . . . . . . 9 (Id‘𝐶) = (Id‘𝐶)
123ad2antrr 723 . . . . . . . . 9 (((𝜑𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂𝐵) → 𝐶 ∈ Cat)
13 simpr 485 . . . . . . . . 9 (((𝜑𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂𝐵) → 𝑂𝐵)
141, 2, 11, 12, 13catidcl 17402 . . . . . . . 8 (((𝜑𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂𝐵) → ((Id‘𝐶)‘𝑂) ∈ (𝑂𝐻𝑂))
15 fvex 6784 . . . . . . . . . . . . 13 ((Id‘𝐶)‘𝑂) ∈ V
1615elsn 4582 . . . . . . . . . . . 12 (((Id‘𝐶)‘𝑂) ∈ {} ↔ ((Id‘𝐶)‘𝑂) = )
17 eqcom 2747 . . . . . . . . . . . 12 (((Id‘𝐶)‘𝑂) = = ((Id‘𝐶)‘𝑂))
18 sneqbg 4780 . . . . . . . . . . . . . 14 ( ∈ V → ({} = {((Id‘𝐶)‘𝑂)} ↔ = ((Id‘𝐶)‘𝑂)))
1918bicomd 222 . . . . . . . . . . . . 13 ( ∈ V → ( = ((Id‘𝐶)‘𝑂) ↔ {} = {((Id‘𝐶)‘𝑂)}))
2019elv 3437 . . . . . . . . . . . 12 ( = ((Id‘𝐶)‘𝑂) ↔ {} = {((Id‘𝐶)‘𝑂)})
2116, 17, 203bitri 297 . . . . . . . . . . 11 (((Id‘𝐶)‘𝑂) ∈ {} ↔ {} = {((Id‘𝐶)‘𝑂)})
2221biimpi 215 . . . . . . . . . 10 (((Id‘𝐶)‘𝑂) ∈ {} → {} = {((Id‘𝐶)‘𝑂)})
2322a1i 11 . . . . . . . . 9 ((𝑂𝐻𝑂) = {} → (((Id‘𝐶)‘𝑂) ∈ {} → {} = {((Id‘𝐶)‘𝑂)}))
24 eleq2 2829 . . . . . . . . 9 ((𝑂𝐻𝑂) = {} → (((Id‘𝐶)‘𝑂) ∈ (𝑂𝐻𝑂) ↔ ((Id‘𝐶)‘𝑂) ∈ {}))
25 eqeq1 2744 . . . . . . . . 9 ((𝑂𝐻𝑂) = {} → ((𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)} ↔ {} = {((Id‘𝐶)‘𝑂)}))
2623, 24, 253imtr4d 294 . . . . . . . 8 ((𝑂𝐻𝑂) = {} → (((Id‘𝐶)‘𝑂) ∈ (𝑂𝐻𝑂) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
2714, 26syl5 34 . . . . . . 7 ((𝑂𝐻𝑂) = {} → (((𝜑𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂𝐵) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
2827exlimiv 1937 . . . . . 6 (∃(𝑂𝐻𝑂) = {} → (((𝜑𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂𝐵) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
2928com12 32 . . . . 5 (((𝜑𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂𝐵) → (∃(𝑂𝐻𝑂) = {} → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
3010, 29syl5bi 241 . . . 4 (((𝜑𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂𝐵) → (∃! ∈ (𝑂𝐻𝑂) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
319, 30syld 47 . . 3 (((𝜑𝑂 ∈ (TermO‘𝐶)) ∧ 𝑂𝐵) → (∀𝑜𝐵 ∃! ∈ (𝑜𝐻𝑂) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
3231expimpd 454 . 2 ((𝜑𝑂 ∈ (TermO‘𝐶)) → ((𝑂𝐵 ∧ ∀𝑜𝐵 ∃! ∈ (𝑜𝐻𝑂)) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
334, 32mpd 15 1 ((𝜑𝑂 ∈ (TermO‘𝐶)) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1542  wex 1786  wcel 2110  ∃!weu 2570  wral 3066  Vcvv 3431  {csn 4567  cfv 6432  (class class class)co 7272  Basecbs 16923  Hom chom 16984  Catccat 17384  Idccid 17385  TermOctermo 17708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pr 5356
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-riota 7229  df-ov 7275  df-cat 17388  df-cid 17389  df-termo 17711
This theorem is referenced by:  2termoinv  17743
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