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Theorem initoval 17245
Description: The value of the initial object function, i.e. the set of all initial objects of a category. (Contributed by AV, 3-Apr-2020.)
Hypotheses
Ref Expression
initoval.c (𝜑𝐶 ∈ Cat)
initoval.b 𝐵 = (Base‘𝐶)
initoval.h 𝐻 = (Hom ‘𝐶)
Assertion
Ref Expression
initoval (𝜑 → (InitO‘𝐶) = {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑎𝐻𝑏)})
Distinct variable groups:   𝑎,𝑏,   𝐵,𝑎,𝑏   𝐶,𝑎,𝑏,
Allowed substitution hints:   𝜑(,𝑎,𝑏)   𝐵()   𝐻(,𝑎,𝑏)

Proof of Theorem initoval
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 df-inito 17239 . 2 InitO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑎(Hom ‘𝑐)𝑏)})
2 fveq2 6663 . . . 4 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
3 initoval.b . . . 4 𝐵 = (Base‘𝐶)
42, 3syl6eqr 2871 . . 3 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
5 fveq2 6663 . . . . . . . 8 (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
6 initoval.h . . . . . . . 8 𝐻 = (Hom ‘𝐶)
75, 6syl6eqr 2871 . . . . . . 7 (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
87oveqd 7162 . . . . . 6 (𝑐 = 𝐶 → (𝑎(Hom ‘𝑐)𝑏) = (𝑎𝐻𝑏))
98eleq2d 2895 . . . . 5 (𝑐 = 𝐶 → ( ∈ (𝑎(Hom ‘𝑐)𝑏) ↔ ∈ (𝑎𝐻𝑏)))
109eubidv 2665 . . . 4 (𝑐 = 𝐶 → (∃! ∈ (𝑎(Hom ‘𝑐)𝑏) ↔ ∃! ∈ (𝑎𝐻𝑏)))
114, 10raleqbidv 3399 . . 3 (𝑐 = 𝐶 → (∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑎(Hom ‘𝑐)𝑏) ↔ ∀𝑏𝐵 ∃! ∈ (𝑎𝐻𝑏)))
124, 11rabeqbidv 3483 . 2 (𝑐 = 𝐶 → {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑎(Hom ‘𝑐)𝑏)} = {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑎𝐻𝑏)})
13 initoval.c . 2 (𝜑𝐶 ∈ Cat)
143fvexi 6677 . . . 4 𝐵 ∈ V
1514rabex 5226 . . 3 {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑎𝐻𝑏)} ∈ V
1615a1i 11 . 2 (𝜑 → {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑎𝐻𝑏)} ∈ V)
171, 12, 13, 16fvmptd3 6783 1 (𝜑 → (InitO‘𝐶) = {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑎𝐻𝑏)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  ∃!weu 2646  wral 3135  {crab 3139  Vcvv 3492  cfv 6348  (class class class)co 7145  Basecbs 16471  Hom chom 16564  Catccat 16923  InitOcinito 17236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-inito 17239
This theorem is referenced by:  isinito  17248  isinitoi  17251
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