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Theorem isinito 16419
Description: The predicate "is an initial object" of a category. (Contributed by AV, 3-Apr-2020.)
Hypotheses
Ref Expression
isinito.b 𝐵 = (Base‘𝐶)
isinito.h 𝐻 = (Hom ‘𝐶)
isinito.c (𝜑𝐶 ∈ Cat)
isinito.i (𝜑𝐼𝐵)
Assertion
Ref Expression
isinito (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ ∀𝑏𝐵 ∃! ∈ (𝐼𝐻𝑏)))
Distinct variable groups:   𝐵,𝑏   𝐶,𝑏,   𝐼,𝑏,
Allowed substitution hints:   𝜑(,𝑏)   𝐵()   𝐻(,𝑏)

Proof of Theorem isinito
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 isinito.c . . . 4 (𝜑𝐶 ∈ Cat)
2 isinito.b . . . 4 𝐵 = (Base‘𝐶)
3 isinito.h . . . 4 𝐻 = (Hom ‘𝐶)
41, 2, 3initoval 16416 . . 3 (𝜑 → (InitO‘𝐶) = {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑖𝐻𝑏)})
54eleq2d 2672 . 2 (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼 ∈ {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑖𝐻𝑏)}))
6 isinito.i . . 3 (𝜑𝐼𝐵)
7 oveq1 6534 . . . . . . 7 (𝑖 = 𝐼 → (𝑖𝐻𝑏) = (𝐼𝐻𝑏))
87eleq2d 2672 . . . . . 6 (𝑖 = 𝐼 → ( ∈ (𝑖𝐻𝑏) ↔ ∈ (𝐼𝐻𝑏)))
98eubidv 2477 . . . . 5 (𝑖 = 𝐼 → (∃! ∈ (𝑖𝐻𝑏) ↔ ∃! ∈ (𝐼𝐻𝑏)))
109ralbidv 2968 . . . 4 (𝑖 = 𝐼 → (∀𝑏𝐵 ∃! ∈ (𝑖𝐻𝑏) ↔ ∀𝑏𝐵 ∃! ∈ (𝐼𝐻𝑏)))
1110elrab3 3331 . . 3 (𝐼𝐵 → (𝐼 ∈ {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑖𝐻𝑏)} ↔ ∀𝑏𝐵 ∃! ∈ (𝐼𝐻𝑏)))
126, 11syl 17 . 2 (𝜑 → (𝐼 ∈ {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑖𝐻𝑏)} ↔ ∀𝑏𝐵 ∃! ∈ (𝐼𝐻𝑏)))
135, 12bitrd 266 1 (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ ∀𝑏𝐵 ∃! ∈ (𝐼𝐻𝑏)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194   = wceq 1474  wcel 1976  ∃!weu 2457  wral 2895  {crab 2899  cfv 5790  (class class class)co 6527  Basecbs 15641  Hom chom 15725  Catccat 16094  InitOcinito 16407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fv 5798  df-ov 6530  df-inito 16410
This theorem is referenced by:  isinitoi  16422  initoeu2  16435  zrinitorngc  41787  irinitoringc  41856  zrninitoringc  41858
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