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Theorem isinito 17928
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 17925 . . 3 (𝜑 → (InitO‘𝐶) = {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑖𝐻𝑏)})
54eleq2d 2818 . 2 (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼 ∈ {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑖𝐻𝑏)}))
6 isinito.i . . 3 (𝜑𝐼𝐵)
7 oveq1 7400 . . . . . . 7 (𝑖 = 𝐼 → (𝑖𝐻𝑏) = (𝐼𝐻𝑏))
87eleq2d 2818 . . . . . 6 (𝑖 = 𝐼 → ( ∈ (𝑖𝐻𝑏) ↔ ∈ (𝐼𝐻𝑏)))
98eubidv 2579 . . . . 5 (𝑖 = 𝐼 → (∃! ∈ (𝑖𝐻𝑏) ↔ ∃! ∈ (𝐼𝐻𝑏)))
109ralbidv 3176 . . . 4 (𝑖 = 𝐼 → (∀𝑏𝐵 ∃! ∈ (𝑖𝐻𝑏) ↔ ∀𝑏𝐵 ∃! ∈ (𝐼𝐻𝑏)))
1110elrab3 3680 . . 3 (𝐼𝐵 → (𝐼 ∈ {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑖𝐻𝑏)} ↔ ∀𝑏𝐵 ∃! ∈ (𝐼𝐻𝑏)))
126, 11syl 17 . 2 (𝜑 → (𝐼 ∈ {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑖𝐻𝑏)} ↔ ∀𝑏𝐵 ∃! ∈ (𝐼𝐻𝑏)))
135, 12bitrd 278 1 (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ ∀𝑏𝐵 ∃! ∈ (𝐼𝐻𝑏)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  wcel 2106  ∃!weu 2561  wral 3060  {crab 3431  cfv 6532  (class class class)co 7393  Basecbs 17126  Hom chom 17190  Catccat 17590  InitOcinito 17913
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6484  df-fun 6534  df-fv 6540  df-ov 7396  df-inito 17916
This theorem is referenced by:  isinitoi  17931  initoeu2  17948  zrinitorngc  46544  irinitoringc  46613  zrninitoringc  46615
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