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Theorem initoid 16702
 Description: For an initial object, the identity arrow is the one and only morphism of the object to the object itself. (Contributed by AV, 6-Apr-2020.)
Hypotheses
Ref Expression
isinitoi.b 𝐵 = (Base‘𝐶)
isinitoi.h 𝐻 = (Hom ‘𝐶)
isinitoi.c (𝜑𝐶 ∈ Cat)
Assertion
Ref Expression
initoid ((𝜑𝑂 ∈ (InitO‘𝐶)) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)})

Proof of Theorem initoid
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, 3isinitoi 16700 . 2 ((𝜑𝑂 ∈ (InitO‘𝐶)) → (𝑂𝐵 ∧ ∀𝑜𝐵 ∃! ∈ (𝑂𝐻𝑜)))
5 oveq2 6698 . . . . . . . 8 (𝑜 = 𝑂 → (𝑂𝐻𝑜) = (𝑂𝐻𝑂))
65eleq2d 2716 . . . . . . 7 (𝑜 = 𝑂 → ( ∈ (𝑂𝐻𝑜) ↔ ∈ (𝑂𝐻𝑂)))
76eubidv 2518 . . . . . 6 (𝑜 = 𝑂 → (∃! ∈ (𝑂𝐻𝑜) ↔ ∃! ∈ (𝑂𝐻𝑂)))
87rspcv 3336 . . . . 5 (𝑂𝐵 → (∀𝑜𝐵 ∃! ∈ (𝑂𝐻𝑜) → ∃! ∈ (𝑂𝐻𝑂)))
98adantl 481 . . . 4 (((𝜑𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂𝐵) → (∀𝑜𝐵 ∃! ∈ (𝑂𝐻𝑜) → ∃! ∈ (𝑂𝐻𝑂)))
10 eusn 4297 . . . . 5 (∃! ∈ (𝑂𝐻𝑂) ↔ ∃(𝑂𝐻𝑂) = {})
11 eqid 2651 . . . . . . . . 9 (Id‘𝐶) = (Id‘𝐶)
123ad2antrr 762 . . . . . . . . 9 (((𝜑𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂𝐵) → 𝐶 ∈ Cat)
13 simpr 476 . . . . . . . . 9 (((𝜑𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂𝐵) → 𝑂𝐵)
141, 2, 11, 12, 13catidcl 16390 . . . . . . . 8 (((𝜑𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂𝐵) → ((Id‘𝐶)‘𝑂) ∈ (𝑂𝐻𝑂))
15 fvex 6239 . . . . . . . . . . . . 13 ((Id‘𝐶)‘𝑂) ∈ V
1615elsn 4225 . . . . . . . . . . . 12 (((Id‘𝐶)‘𝑂) ∈ {} ↔ ((Id‘𝐶)‘𝑂) = )
17 eqcom 2658 . . . . . . . . . . . 12 (((Id‘𝐶)‘𝑂) = = ((Id‘𝐶)‘𝑂))
18 vex 3234 . . . . . . . . . . . . 13 ∈ V
19 sneqbg 4406 . . . . . . . . . . . . . 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 2719 . . . . . . . . 9 ((𝑂𝐻𝑂) = {} → (((Id‘𝐶)‘𝑂) ∈ (𝑂𝐻𝑂) ↔ ((Id‘𝐶)‘𝑂) ∈ {}))
26 eqeq1 2655 . . . . . . . . 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 ∈ (InitO‘𝐶)) ∧ 𝑂𝐵) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
2928exlimiv 1898 . . . . . 6 (∃(𝑂𝐻𝑂) = {} → (((𝜑𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂𝐵) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
3029com12 32 . . . . 5 (((𝜑𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂𝐵) → (∃(𝑂𝐻𝑂) = {} → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
3110, 30syl5bi 232 . . . 4 (((𝜑𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂𝐵) → (∃! ∈ (𝑂𝐻𝑂) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
329, 31syld 47 . . 3 (((𝜑𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂𝐵) → (∀𝑜𝐵 ∃! ∈ (𝑂𝐻𝑜) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
3332expimpd 628 . 2 ((𝜑𝑂 ∈ (InitO‘𝐶)) → ((𝑂𝐵 ∧ ∀𝑜𝐵 ∃! ∈ (𝑂𝐻𝑜)) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
344, 33mpd 15 1 ((𝜑𝑂 ∈ (InitO‘𝐶)) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523  ∃wex 1744   ∈ wcel 2030  ∃!weu 2498  ∀wral 2941  Vcvv 3231  {csn 4210  ‘cfv 5926  (class class class)co 6690  Basecbs 15904  Hom chom 15999  Catccat 16372  Idccid 16373  InitOcinito 16685 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-cat 16376  df-cid 16377  df-inito 16688 This theorem is referenced by:  2initoinv  16707
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