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Theorem initoval 16847
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 16841 . . 3 InitO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑎(Hom ‘𝑐)𝑏)})
21a1i 11 . 2 (𝜑 → InitO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑎(Hom ‘𝑐)𝑏)}))
3 fveq2 6404 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
4 initoval.b . . . . 5 𝐵 = (Base‘𝐶)
53, 4syl6eqr 2858 . . . 4 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
6 fveq2 6404 . . . . . . . . 9 (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
7 initoval.h . . . . . . . . 9 𝐻 = (Hom ‘𝐶)
86, 7syl6eqr 2858 . . . . . . . 8 (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
98oveqd 6887 . . . . . . 7 (𝑐 = 𝐶 → (𝑎(Hom ‘𝑐)𝑏) = (𝑎𝐻𝑏))
109eleq2d 2871 . . . . . 6 (𝑐 = 𝐶 → ( ∈ (𝑎(Hom ‘𝑐)𝑏) ↔ ∈ (𝑎𝐻𝑏)))
1110eubidv 2636 . . . . 5 (𝑐 = 𝐶 → (∃! ∈ (𝑎(Hom ‘𝑐)𝑏) ↔ ∃! ∈ (𝑎𝐻𝑏)))
125, 11raleqbidv 3341 . . . 4 (𝑐 = 𝐶 → (∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑎(Hom ‘𝑐)𝑏) ↔ ∀𝑏𝐵 ∃! ∈ (𝑎𝐻𝑏)))
135, 12rabeqbidv 3385 . . 3 (𝑐 = 𝐶 → {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑎(Hom ‘𝑐)𝑏)} = {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑎𝐻𝑏)})
1413adantl 469 . 2 ((𝜑𝑐 = 𝐶) → {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑎(Hom ‘𝑐)𝑏)} = {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑎𝐻𝑏)})
15 initoval.c . 2 (𝜑𝐶 ∈ Cat)
164fvexi 6418 . . . 4 𝐵 ∈ V
1716rabex 5007 . . 3 {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑎𝐻𝑏)} ∈ V
1817a1i 11 . 2 (𝜑 → {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑎𝐻𝑏)} ∈ V)
192, 14, 15, 18fvmptd 6505 1 (𝜑 → (InitO‘𝐶) = {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑎𝐻𝑏)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1637  wcel 2156  ∃!weu 2630  wral 3096  {crab 3100  Vcvv 3391  cmpt 4923  cfv 6097  (class class class)co 6870  Basecbs 16064  Hom chom 16160  Catccat 16525  InitOcinito 16838
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 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096
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 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-iota 6060  df-fun 6099  df-fv 6105  df-ov 6873  df-inito 16841
This theorem is referenced by:  isinito  16850  isinitoi  16853
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