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Theorem initoval 16854
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 16848 . . 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 6333 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
4 initoval.b . . . . 5 𝐵 = (Base‘𝐶)
53, 4syl6eqr 2823 . . . 4 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
6 fveq2 6333 . . . . . . . . 9 (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
7 initoval.h . . . . . . . . 9 𝐻 = (Hom ‘𝐶)
86, 7syl6eqr 2823 . . . . . . . 8 (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
98oveqd 6813 . . . . . . 7 (𝑐 = 𝐶 → (𝑎(Hom ‘𝑐)𝑏) = (𝑎𝐻𝑏))
109eleq2d 2836 . . . . . 6 (𝑐 = 𝐶 → ( ∈ (𝑎(Hom ‘𝑐)𝑏) ↔ ∈ (𝑎𝐻𝑏)))
1110eubidv 2638 . . . . 5 (𝑐 = 𝐶 → (∃! ∈ (𝑎(Hom ‘𝑐)𝑏) ↔ ∃! ∈ (𝑎𝐻𝑏)))
125, 11raleqbidv 3301 . . . 4 (𝑐 = 𝐶 → (∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑎(Hom ‘𝑐)𝑏) ↔ ∀𝑏𝐵 ∃! ∈ (𝑎𝐻𝑏)))
135, 12rabeqbidv 3345 . . 3 (𝑐 = 𝐶 → {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑎(Hom ‘𝑐)𝑏)} = {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑎𝐻𝑏)})
1413adantl 467 . 2 ((𝜑𝑐 = 𝐶) → {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑎(Hom ‘𝑐)𝑏)} = {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑎𝐻𝑏)})
15 initoval.c . 2 (𝜑𝐶 ∈ Cat)
164fvexi 6345 . . . 4 𝐵 ∈ V
1716rabex 4947 . . 3 {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑎𝐻𝑏)} ∈ V
1817a1i 11 . 2 (𝜑 → {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑎𝐻𝑏)} ∈ V)
192, 14, 15, 18fvmptd 6432 1 (𝜑 → (InitO‘𝐶) = {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑎𝐻𝑏)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  ∃!weu 2618  wral 3061  {crab 3065  Vcvv 3351  cmpt 4864  cfv 6030  (class class class)co 6796  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 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  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 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fv 6038  df-ov 6799  df-inito 16848
This theorem is referenced by:  isinito  16857  isinitoi  16860
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