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Theorem isinito 17093
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 17090 . . 3 (𝜑 → (InitO‘𝐶) = {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑖𝐻𝑏)})
54eleq2d 2870 . 2 (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼 ∈ {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑖𝐻𝑏)}))
6 isinito.i . . 3 (𝜑𝐼𝐵)
7 oveq1 7030 . . . . . . 7 (𝑖 = 𝐼 → (𝑖𝐻𝑏) = (𝐼𝐻𝑏))
87eleq2d 2870 . . . . . 6 (𝑖 = 𝐼 → ( ∈ (𝑖𝐻𝑏) ↔ ∈ (𝐼𝐻𝑏)))
98eubidv 2634 . . . . 5 (𝑖 = 𝐼 → (∃! ∈ (𝑖𝐻𝑏) ↔ ∃! ∈ (𝐼𝐻𝑏)))
109ralbidv 3166 . . . 4 (𝑖 = 𝐼 → (∀𝑏𝐵 ∃! ∈ (𝑖𝐻𝑏) ↔ ∀𝑏𝐵 ∃! ∈ (𝐼𝐻𝑏)))
1110elrab3 3622 . . 3 (𝐼𝐵 → (𝐼 ∈ {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑖𝐻𝑏)} ↔ ∀𝑏𝐵 ∃! ∈ (𝐼𝐻𝑏)))
126, 11syl 17 . 2 (𝜑 → (𝐼 ∈ {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑖𝐻𝑏)} ↔ ∀𝑏𝐵 ∃! ∈ (𝐼𝐻𝑏)))
135, 12bitrd 280 1 (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ ∀𝑏𝐵 ∃! ∈ (𝐼𝐻𝑏)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1525  wcel 2083  ∃!weu 2613  wral 3107  {crab 3111  cfv 6232  (class class class)co 7023  Basecbs 16316  Hom chom 16409  Catccat 16768  InitOcinito 17081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-iota 6196  df-fun 6234  df-fv 6240  df-ov 7026  df-inito 17084
This theorem is referenced by:  isinitoi  17096  initoeu2  17109  zrinitorngc  43771  irinitoringc  43840  zrninitoringc  43842
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