Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  initoval Structured version   Visualization version   GIF version

Theorem initoval 17266
 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 17260 . 2 InitO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑎(Hom ‘𝑐)𝑏)})
2 fveq2 6652 . . . 4 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
3 initoval.b . . . 4 𝐵 = (Base‘𝐶)
42, 3eqtr4di 2851 . . 3 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
5 fveq2 6652 . . . . . . . 8 (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
6 initoval.h . . . . . . . 8 𝐻 = (Hom ‘𝐶)
75, 6eqtr4di 2851 . . . . . . 7 (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
87oveqd 7159 . . . . . 6 (𝑐 = 𝐶 → (𝑎(Hom ‘𝑐)𝑏) = (𝑎𝐻𝑏))
98eleq2d 2875 . . . . 5 (𝑐 = 𝐶 → ( ∈ (𝑎(Hom ‘𝑐)𝑏) ↔ ∈ (𝑎𝐻𝑏)))
109eubidv 2647 . . . 4 (𝑐 = 𝐶 → (∃! ∈ (𝑎(Hom ‘𝑐)𝑏) ↔ ∃! ∈ (𝑎𝐻𝑏)))
114, 10raleqbidv 3354 . . 3 (𝑐 = 𝐶 → (∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑎(Hom ‘𝑐)𝑏) ↔ ∀𝑏𝐵 ∃! ∈ (𝑎𝐻𝑏)))
124, 11rabeqbidv 3433 . 2 (𝑐 = 𝐶 → {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑎(Hom ‘𝑐)𝑏)} = {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑎𝐻𝑏)})
13 initoval.c . 2 (𝜑𝐶 ∈ Cat)
143fvexi 6666 . . . 4 𝐵 ∈ V
1514rabex 5202 . . 3 {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑎𝐻𝑏)} ∈ V
1615a1i 11 . 2 (𝜑 → {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑎𝐻𝑏)} ∈ V)
171, 12, 13, 16fvmptd3 6775 1 (𝜑 → (InitO‘𝐶) = {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑎𝐻𝑏)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  ∃!weu 2628  ∀wral 3106  {crab 3110  Vcvv 3441  ‘cfv 6329  (class class class)co 7142  Basecbs 16492  Hom chom 16585  Catccat 16944  InitOcinito 17257 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3722  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-nul 4246  df-if 4428  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-iota 6288  df-fun 6331  df-fv 6337  df-ov 7145  df-inito 17260 This theorem is referenced by:  isinito  17269  isinitoi  17272
 Copyright terms: Public domain W3C validator