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Theorem isinitoi 18045
Description: Implication of a class being an initial object. (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
isinitoi ((𝜑𝑂 ∈ (InitO‘𝐶)) → (𝑂𝐵 ∧ ∀𝑏𝐵 ∃! ∈ (𝑂𝐻𝑏)))
Distinct variable groups:   𝐵,𝑏   𝐶,𝑏,   𝑂,𝑏,
Allowed substitution hints:   𝜑(,𝑏)   𝐵()   𝐻(,𝑏)

Proof of Theorem isinitoi
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 isinitoi.c . . . . . 6 (𝜑𝐶 ∈ Cat)
2 isinitoi.b . . . . . 6 𝐵 = (Base‘𝐶)
3 isinitoi.h . . . . . 6 𝐻 = (Hom ‘𝐶)
41, 2, 3initoval 18039 . . . . 5 (𝜑 → (InitO‘𝐶) = {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑎𝐻𝑏)})
54eleq2d 2826 . . . 4 (𝜑 → (𝑂 ∈ (InitO‘𝐶) ↔ 𝑂 ∈ {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑎𝐻𝑏)}))
6 elrabi 3686 . . . 4 (𝑂 ∈ {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑎𝐻𝑏)} → 𝑂𝐵)
75, 6biimtrdi 253 . . 3 (𝜑 → (𝑂 ∈ (InitO‘𝐶) → 𝑂𝐵))
87imp 406 . 2 ((𝜑𝑂 ∈ (InitO‘𝐶)) → 𝑂𝐵)
91adantr 480 . . . . 5 ((𝜑𝑂𝐵) → 𝐶 ∈ Cat)
10 simpr 484 . . . . 5 ((𝜑𝑂𝐵) → 𝑂𝐵)
112, 3, 9, 10isinito 18042 . . . 4 ((𝜑𝑂𝐵) → (𝑂 ∈ (InitO‘𝐶) ↔ ∀𝑏𝐵 ∃! ∈ (𝑂𝐻𝑏)))
1211biimpd 229 . . 3 ((𝜑𝑂𝐵) → (𝑂 ∈ (InitO‘𝐶) → ∀𝑏𝐵 ∃! ∈ (𝑂𝐻𝑏)))
1312impancom 451 . 2 ((𝜑𝑂 ∈ (InitO‘𝐶)) → (𝑂𝐵 → ∀𝑏𝐵 ∃! ∈ (𝑂𝐻𝑏)))
148, 13jcai 516 1 ((𝜑𝑂 ∈ (InitO‘𝐶)) → (𝑂𝐵 ∧ ∀𝑏𝐵 ∃! ∈ (𝑂𝐻𝑏)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  ∃!weu 2567  wral 3060  {crab 3435  cfv 6560  (class class class)co 7432  Basecbs 17248  Hom chom 17309  Catccat 17708  InitOcinito 18027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-inito 18030
This theorem is referenced by:  initoid  18047  initoo  18053  initoeu1  18057  initoeu2  18062
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