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Theorem initoid 17531
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 17529 . 2 ((𝜑𝑂 ∈ (InitO‘𝐶)) → (𝑂𝐵 ∧ ∀𝑜𝐵 ∃! ∈ (𝑂𝐻𝑜)))
5 oveq2 7239 . . . . . . . 8 (𝑜 = 𝑂 → (𝑂𝐻𝑜) = (𝑂𝐻𝑂))
65eleq2d 2824 . . . . . . 7 (𝑜 = 𝑂 → ( ∈ (𝑂𝐻𝑜) ↔ ∈ (𝑂𝐻𝑂)))
76eubidv 2586 . . . . . 6 (𝑜 = 𝑂 → (∃! ∈ (𝑂𝐻𝑜) ↔ ∃! ∈ (𝑂𝐻𝑂)))
87rspcv 3544 . . . . 5 (𝑂𝐵 → (∀𝑜𝐵 ∃! ∈ (𝑂𝐻𝑜) → ∃! ∈ (𝑂𝐻𝑂)))
98adantl 485 . . . 4 (((𝜑𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂𝐵) → (∀𝑜𝐵 ∃! ∈ (𝑂𝐻𝑜) → ∃! ∈ (𝑂𝐻𝑂)))
10 eusn 4660 . . . . 5 (∃! ∈ (𝑂𝐻𝑂) ↔ ∃(𝑂𝐻𝑂) = {})
11 eqid 2738 . . . . . . . . 9 (Id‘𝐶) = (Id‘𝐶)
123ad2antrr 726 . . . . . . . . 9 (((𝜑𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂𝐵) → 𝐶 ∈ Cat)
13 simpr 488 . . . . . . . . 9 (((𝜑𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂𝐵) → 𝑂𝐵)
141, 2, 11, 12, 13catidcl 17209 . . . . . . . 8 (((𝜑𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂𝐵) → ((Id‘𝐶)‘𝑂) ∈ (𝑂𝐻𝑂))
15 fvex 6748 . . . . . . . . . . . . 13 ((Id‘𝐶)‘𝑂) ∈ V
1615elsn 4570 . . . . . . . . . . . 12 (((Id‘𝐶)‘𝑂) ∈ {} ↔ ((Id‘𝐶)‘𝑂) = )
17 eqcom 2745 . . . . . . . . . . . 12 (((Id‘𝐶)‘𝑂) = = ((Id‘𝐶)‘𝑂))
18 sneqbg 4768 . . . . . . . . . . . . . 14 ( ∈ V → ({} = {((Id‘𝐶)‘𝑂)} ↔ = ((Id‘𝐶)‘𝑂)))
1918bicomd 226 . . . . . . . . . . . . 13 ( ∈ V → ( = ((Id‘𝐶)‘𝑂) ↔ {} = {((Id‘𝐶)‘𝑂)}))
2019elv 3426 . . . . . . . . . . . 12 ( = ((Id‘𝐶)‘𝑂) ↔ {} = {((Id‘𝐶)‘𝑂)})
2116, 17, 203bitri 300 . . . . . . . . . . 11 (((Id‘𝐶)‘𝑂) ∈ {} ↔ {} = {((Id‘𝐶)‘𝑂)})
2221biimpi 219 . . . . . . . . . 10 (((Id‘𝐶)‘𝑂) ∈ {} → {} = {((Id‘𝐶)‘𝑂)})
2322a1i 11 . . . . . . . . 9 ((𝑂𝐻𝑂) = {} → (((Id‘𝐶)‘𝑂) ∈ {} → {} = {((Id‘𝐶)‘𝑂)}))
24 eleq2 2827 . . . . . . . . 9 ((𝑂𝐻𝑂) = {} → (((Id‘𝐶)‘𝑂) ∈ (𝑂𝐻𝑂) ↔ ((Id‘𝐶)‘𝑂) ∈ {}))
25 eqeq1 2742 . . . . . . . . 9 ((𝑂𝐻𝑂) = {} → ((𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)} ↔ {} = {((Id‘𝐶)‘𝑂)}))
2623, 24, 253imtr4d 297 . . . . . . . 8 ((𝑂𝐻𝑂) = {} → (((Id‘𝐶)‘𝑂) ∈ (𝑂𝐻𝑂) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
2714, 26syl5 34 . . . . . . 7 ((𝑂𝐻𝑂) = {} → (((𝜑𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂𝐵) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
2827exlimiv 1938 . . . . . 6 (∃(𝑂𝐻𝑂) = {} → (((𝜑𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂𝐵) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
2928com12 32 . . . . 5 (((𝜑𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂𝐵) → (∃(𝑂𝐻𝑂) = {} → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
3010, 29syl5bi 245 . . . 4 (((𝜑𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂𝐵) → (∃! ∈ (𝑂𝐻𝑂) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
319, 30syld 47 . . 3 (((𝜑𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂𝐵) → (∀𝑜𝐵 ∃! ∈ (𝑂𝐻𝑜) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
3231expimpd 457 . 2 ((𝜑𝑂 ∈ (InitO‘𝐶)) → ((𝑂𝐵 ∧ ∀𝑜𝐵 ∃! ∈ (𝑂𝐻𝑜)) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
334, 32mpd 15 1 ((𝜑𝑂 ∈ (InitO‘𝐶)) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wex 1787  wcel 2111  ∃!weu 2568  wral 3062  Vcvv 3420  {csn 4555  cfv 6397  (class class class)co 7231  Basecbs 16784  Hom chom 16837  Catccat 17191  Idccid 17192  InitOcinito 17511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pr 5336
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-reu 3069  df-rmo 3070  df-rab 3071  df-v 3422  df-sbc 3709  df-csb 3826  df-dif 3883  df-un 3885  df-in 3887  df-ss 3897  df-nul 4252  df-if 4454  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4834  df-iun 4920  df-br 5068  df-opab 5130  df-mpt 5150  df-id 5469  df-xp 5571  df-rel 5572  df-cnv 5573  df-co 5574  df-dm 5575  df-rn 5576  df-res 5577  df-ima 5578  df-iota 6355  df-fun 6399  df-fn 6400  df-f 6401  df-f1 6402  df-fo 6403  df-f1o 6404  df-fv 6405  df-riota 7188  df-ov 7234  df-cat 17195  df-cid 17196  df-inito 17514
This theorem is referenced by:  2initoinv  17540
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