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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 2836 . 2 (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼 ∈ {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑖𝐻𝑏)}))
6 isinito.i . . 3 (𝜑𝐼𝐵)
7 oveq1 6800 . . . . . . 7 (𝑖 = 𝐼 → (𝑖𝐻𝑏) = (𝐼𝐻𝑏))
87eleq2d 2836 . . . . . 6 (𝑖 = 𝐼 → ( ∈ (𝑖𝐻𝑏) ↔ ∈ (𝐼𝐻𝑏)))
98eubidv 2638 . . . . 5 (𝑖 = 𝐼 → (∃! ∈ (𝑖𝐻𝑏) ↔ ∃! ∈ (𝐼𝐻𝑏)))
109ralbidv 3135 . . . 4 (𝑖 = 𝐼 → (∀𝑏𝐵 ∃! ∈ (𝑖𝐻𝑏) ↔ ∀𝑏𝐵 ∃! ∈ (𝐼𝐻𝑏)))
1110elrab3 3516 . . 3 (𝐼𝐵 → (𝐼 ∈ {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑖𝐻𝑏)} ↔ ∀𝑏𝐵 ∃! ∈ (𝐼𝐻𝑏)))
126, 11syl 17 . 2 (𝜑 → (𝐼 ∈ {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑖𝐻𝑏)} ↔ ∀𝑏𝐵 ∃! ∈ (𝐼𝐻𝑏)))
135, 12bitrd 268 1 (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ ∀𝑏𝐵 ∃! ∈ (𝐼𝐻𝑏)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  wcel 2145  ∃!weu 2618  wral 3061  {crab 3065  cfv 6031  (class class class)co 6793  Basecbs 16064  Hom chom 16160  Catccat 16532  InitOcinito 16845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-inito 16848
This theorem is referenced by:  isinitoi  16860  initoeu2  16873  zrinitorngc  42528  irinitoringc  42597  zrninitoringc  42599
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