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Theorem isinito 16857
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 16854 . . 3 (𝜑 → (InitO‘𝐶) = {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑖𝐻𝑏)})
54eleq2d 2878 . 2 (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼 ∈ {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑖𝐻𝑏)}))
6 isinito.i . . 3 (𝜑𝐼𝐵)
7 oveq1 6884 . . . . . . 7 (𝑖 = 𝐼 → (𝑖𝐻𝑏) = (𝐼𝐻𝑏))
87eleq2d 2878 . . . . . 6 (𝑖 = 𝐼 → ( ∈ (𝑖𝐻𝑏) ↔ ∈ (𝐼𝐻𝑏)))
98eubidv 2643 . . . . 5 (𝑖 = 𝐼 → (∃! ∈ (𝑖𝐻𝑏) ↔ ∃! ∈ (𝐼𝐻𝑏)))
109ralbidv 3181 . . . 4 (𝑖 = 𝐼 → (∀𝑏𝐵 ∃! ∈ (𝑖𝐻𝑏) ↔ ∀𝑏𝐵 ∃! ∈ (𝐼𝐻𝑏)))
1110elrab3 3567 . . 3 (𝐼𝐵 → (𝐼 ∈ {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑖𝐻𝑏)} ↔ ∀𝑏𝐵 ∃! ∈ (𝐼𝐻𝑏)))
126, 11syl 17 . 2 (𝜑 → (𝐼 ∈ {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑖𝐻𝑏)} ↔ ∀𝑏𝐵 ∃! ∈ (𝐼𝐻𝑏)))
135, 12bitrd 270 1 (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ ∀𝑏𝐵 ∃! ∈ (𝐼𝐻𝑏)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197   = wceq 1637  wcel 2157  ∃!weu 2637  wral 3103  {crab 3107  cfv 6104  (class class class)co 6877  Basecbs 16071  Hom chom 16167  Catccat 16532  InitOcinito 16845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-sep 4982  ax-nul 4990  ax-pr 5103
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-if 4287  df-sn 4378  df-pr 4380  df-op 4384  df-uni 4638  df-br 4852  df-opab 4914  df-mpt 4931  df-id 5226  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-iota 6067  df-fun 6106  df-fv 6112  df-ov 6880  df-inito 16848
This theorem is referenced by:  isinitoi  16860  initoeu2  16873  zrinitorngc  42569  irinitoringc  42638  zrninitoringc  42640
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