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Theorem isinitoi 17253
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 17247 . . . . 5 (𝜑 → (InitO‘𝐶) = {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑎𝐻𝑏)})
54eleq2d 2903 . . . 4 (𝜑 → (𝑂 ∈ (InitO‘𝐶) ↔ 𝑂 ∈ {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑎𝐻𝑏)}))
6 elrabi 3679 . . . 4 (𝑂 ∈ {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑎𝐻𝑏)} → 𝑂𝐵)
75, 6syl6bi 254 . . 3 (𝜑 → (𝑂 ∈ (InitO‘𝐶) → 𝑂𝐵))
87imp 407 . 2 ((𝜑𝑂 ∈ (InitO‘𝐶)) → 𝑂𝐵)
91adantr 481 . . . . 5 ((𝜑𝑂𝐵) → 𝐶 ∈ Cat)
10 simpr 485 . . . . 5 ((𝜑𝑂𝐵) → 𝑂𝐵)
112, 3, 9, 10isinito 17250 . . . 4 ((𝜑𝑂𝐵) → (𝑂 ∈ (InitO‘𝐶) ↔ ∀𝑏𝐵 ∃! ∈ (𝑂𝐻𝑏)))
1211biimpd 230 . . 3 ((𝜑𝑂𝐵) → (𝑂 ∈ (InitO‘𝐶) → ∀𝑏𝐵 ∃! ∈ (𝑂𝐻𝑏)))
1312impancom 452 . 2 ((𝜑𝑂 ∈ (InitO‘𝐶)) → (𝑂𝐵 → ∀𝑏𝐵 ∃! ∈ (𝑂𝐻𝑏)))
148, 13jcai 517 1 ((𝜑𝑂 ∈ (InitO‘𝐶)) → (𝑂𝐵 ∧ ∀𝑏𝐵 ∃! ∈ (𝑂𝐻𝑏)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  ∃!weu 2651  wral 3143  {crab 3147  cfv 6352  (class class class)co 7148  Basecbs 16473  Hom chom 16566  Catccat 16925  InitOcinito 17238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6312  df-fun 6354  df-fv 6360  df-ov 7151  df-inito 17241
This theorem is referenced by:  initoid  17255  initoo  17257  initoeu1  17261  initoeu2  17266
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