Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  isinitoi Structured version   Visualization version   GIF version

Theorem isinitoi 17339
 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 17333 . . . . 5 (𝜑 → (InitO‘𝐶) = {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑎𝐻𝑏)})
54eleq2d 2837 . . . 4 (𝜑 → (𝑂 ∈ (InitO‘𝐶) ↔ 𝑂 ∈ {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑎𝐻𝑏)}))
6 elrabi 3598 . . . 4 (𝑂 ∈ {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑎𝐻𝑏)} → 𝑂𝐵)
75, 6syl6bi 256 . . 3 (𝜑 → (𝑂 ∈ (InitO‘𝐶) → 𝑂𝐵))
87imp 410 . 2 ((𝜑𝑂 ∈ (InitO‘𝐶)) → 𝑂𝐵)
91adantr 484 . . . . 5 ((𝜑𝑂𝐵) → 𝐶 ∈ Cat)
10 simpr 488 . . . . 5 ((𝜑𝑂𝐵) → 𝑂𝐵)
112, 3, 9, 10isinito 17336 . . . 4 ((𝜑𝑂𝐵) → (𝑂 ∈ (InitO‘𝐶) ↔ ∀𝑏𝐵 ∃! ∈ (𝑂𝐻𝑏)))
1211biimpd 232 . . 3 ((𝜑𝑂𝐵) → (𝑂 ∈ (InitO‘𝐶) → ∀𝑏𝐵 ∃! ∈ (𝑂𝐻𝑏)))
1312impancom 455 . 2 ((𝜑𝑂 ∈ (InitO‘𝐶)) → (𝑂𝐵 → ∀𝑏𝐵 ∃! ∈ (𝑂𝐻𝑏)))
148, 13jcai 520 1 ((𝜑𝑂 ∈ (InitO‘𝐶)) → (𝑂𝐵 ∧ ∀𝑏𝐵 ∃! ∈ (𝑂𝐻𝑏)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃!weu 2587  ∀wral 3070  {crab 3074  ‘cfv 6340  (class class class)co 7156  Basecbs 16555  Hom chom 16648  Catccat 17007  InitOcinito 17321 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6299  df-fun 6342  df-fv 6348  df-ov 7159  df-inito 17324 This theorem is referenced by:  initoid  17341  initoo  17347  initoeu1  17351  initoeu2  17356
 Copyright terms: Public domain W3C validator