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Theorem initoid 17253
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 17251 . 2 ((𝜑𝑂 ∈ (InitO‘𝐶)) → (𝑂𝐵 ∧ ∀𝑜𝐵 ∃! ∈ (𝑂𝐻𝑜)))
5 oveq2 7153 . . . . . . . 8 (𝑜 = 𝑂 → (𝑂𝐻𝑜) = (𝑂𝐻𝑂))
65eleq2d 2895 . . . . . . 7 (𝑜 = 𝑂 → ( ∈ (𝑂𝐻𝑜) ↔ ∈ (𝑂𝐻𝑂)))
76eubidv 2665 . . . . . 6 (𝑜 = 𝑂 → (∃! ∈ (𝑂𝐻𝑜) ↔ ∃! ∈ (𝑂𝐻𝑂)))
87rspcv 3615 . . . . 5 (𝑂𝐵 → (∀𝑜𝐵 ∃! ∈ (𝑂𝐻𝑜) → ∃! ∈ (𝑂𝐻𝑂)))
98adantl 482 . . . 4 (((𝜑𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂𝐵) → (∀𝑜𝐵 ∃! ∈ (𝑂𝐻𝑜) → ∃! ∈ (𝑂𝐻𝑂)))
10 eusn 4658 . . . . 5 (∃! ∈ (𝑂𝐻𝑂) ↔ ∃(𝑂𝐻𝑂) = {})
11 eqid 2818 . . . . . . . . 9 (Id‘𝐶) = (Id‘𝐶)
123ad2antrr 722 . . . . . . . . 9 (((𝜑𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂𝐵) → 𝐶 ∈ Cat)
13 simpr 485 . . . . . . . . 9 (((𝜑𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂𝐵) → 𝑂𝐵)
141, 2, 11, 12, 13catidcl 16941 . . . . . . . 8 (((𝜑𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂𝐵) → ((Id‘𝐶)‘𝑂) ∈ (𝑂𝐻𝑂))
15 fvex 6676 . . . . . . . . . . . . 13 ((Id‘𝐶)‘𝑂) ∈ V
1615elsn 4572 . . . . . . . . . . . 12 (((Id‘𝐶)‘𝑂) ∈ {} ↔ ((Id‘𝐶)‘𝑂) = )
17 eqcom 2825 . . . . . . . . . . . 12 (((Id‘𝐶)‘𝑂) = = ((Id‘𝐶)‘𝑂))
18 sneqbg 4766 . . . . . . . . . . . . . 14 ( ∈ V → ({} = {((Id‘𝐶)‘𝑂)} ↔ = ((Id‘𝐶)‘𝑂)))
1918bicomd 224 . . . . . . . . . . . . 13 ( ∈ V → ( = ((Id‘𝐶)‘𝑂) ↔ {} = {((Id‘𝐶)‘𝑂)}))
2019elv 3497 . . . . . . . . . . . 12 ( = ((Id‘𝐶)‘𝑂) ↔ {} = {((Id‘𝐶)‘𝑂)})
2116, 17, 203bitri 298 . . . . . . . . . . 11 (((Id‘𝐶)‘𝑂) ∈ {} ↔ {} = {((Id‘𝐶)‘𝑂)})
2221biimpi 217 . . . . . . . . . 10 (((Id‘𝐶)‘𝑂) ∈ {} → {} = {((Id‘𝐶)‘𝑂)})
2322a1i 11 . . . . . . . . 9 ((𝑂𝐻𝑂) = {} → (((Id‘𝐶)‘𝑂) ∈ {} → {} = {((Id‘𝐶)‘𝑂)}))
24 eleq2 2898 . . . . . . . . 9 ((𝑂𝐻𝑂) = {} → (((Id‘𝐶)‘𝑂) ∈ (𝑂𝐻𝑂) ↔ ((Id‘𝐶)‘𝑂) ∈ {}))
25 eqeq1 2822 . . . . . . . . 9 ((𝑂𝐻𝑂) = {} → ((𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)} ↔ {} = {((Id‘𝐶)‘𝑂)}))
2623, 24, 253imtr4d 295 . . . . . . . 8 ((𝑂𝐻𝑂) = {} → (((Id‘𝐶)‘𝑂) ∈ (𝑂𝐻𝑂) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
2714, 26syl5 34 . . . . . . 7 ((𝑂𝐻𝑂) = {} → (((𝜑𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂𝐵) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
2827exlimiv 1922 . . . . . 6 (∃(𝑂𝐻𝑂) = {} → (((𝜑𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂𝐵) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
2928com12 32 . . . . 5 (((𝜑𝑂 ∈ (InitO‘𝐶)) ∧ 𝑂𝐵) → (∃(𝑂𝐻𝑂) = {} → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)}))
3010, 29syl5bi 243 . . . 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 454 . 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 207  wa 396   = wceq 1528  wex 1771  wcel 2105  ∃!weu 2646  wral 3135  Vcvv 3492  {csn 4557  cfv 6348  (class class class)co 7145  Basecbs 16471  Hom chom 16564  Catccat 16923  Idccid 16924  InitOcinito 17236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-cat 16927  df-cid 16928  df-inito 17239
This theorem is referenced by:  2initoinv  17258
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