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Theorem termoval 16471
 Description: The value of the terminal object function, i.e. the set of all terminal 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
termoval (𝜑 → (TermO‘𝐶) = {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑎)})
Distinct variable groups:   𝑎,𝑏,   𝐵,𝑎,𝑏   𝐶,𝑎,𝑏,
Allowed substitution hints:   𝜑(,𝑎,𝑏)   𝐵()   𝐻(,𝑎,𝑏)

Proof of Theorem termoval
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 df-termo 16465 . . 3 TermO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑏(Hom ‘𝑐)𝑎)})
21a1i 11 . 2 (𝜑 → TermO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑏(Hom ‘𝑐)𝑎)}))
3 fveq2 6103 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
4 initoval.b . . . . 5 𝐵 = (Base‘𝐶)
53, 4syl6eqr 2662 . . . 4 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
6 fveq2 6103 . . . . . . . . 9 (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
7 initoval.h . . . . . . . . 9 𝐻 = (Hom ‘𝐶)
86, 7syl6eqr 2662 . . . . . . . 8 (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
98oveqd 6566 . . . . . . 7 (𝑐 = 𝐶 → (𝑏(Hom ‘𝑐)𝑎) = (𝑏𝐻𝑎))
109eleq2d 2673 . . . . . 6 (𝑐 = 𝐶 → ( ∈ (𝑏(Hom ‘𝑐)𝑎) ↔ ∈ (𝑏𝐻𝑎)))
1110eubidv 2478 . . . . 5 (𝑐 = 𝐶 → (∃! ∈ (𝑏(Hom ‘𝑐)𝑎) ↔ ∃! ∈ (𝑏𝐻𝑎)))
125, 11raleqbidv 3129 . . . 4 (𝑐 = 𝐶 → (∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑏(Hom ‘𝑐)𝑎) ↔ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑎)))
135, 12rabeqbidv 3168 . . 3 (𝑐 = 𝐶 → {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑏(Hom ‘𝑐)𝑎)} = {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑎)})
1413adantl 481 . 2 ((𝜑𝑐 = 𝐶) → {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑏(Hom ‘𝑐)𝑎)} = {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑎)})
15 initoval.c . 2 (𝜑𝐶 ∈ Cat)
16 fvex 6113 . . . . 5 (Base‘𝐶) ∈ V
174, 16eqeltri 2684 . . . 4 𝐵 ∈ V
1817rabex 4740 . . 3 {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑎)} ∈ V
1918a1i 11 . 2 (𝜑 → {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑎)} ∈ V)
202, 14, 15, 19fvmptd 6197 1 (𝜑 → (TermO‘𝐶) = {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑎)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∃!weu 2458  ∀wral 2896  {crab 2900  Vcvv 3173   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  Hom chom 15779  Catccat 16148  TermOctermo 16462 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-termo 16465 This theorem is referenced by:  istermo  16474  istermoi  16477
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