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Theorem termoval 18048
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 18039 . 2 TermO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑏(Hom ‘𝑐)𝑎)})
2 fveq2 6907 . . . 4 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
3 initoval.b . . . 4 𝐵 = (Base‘𝐶)
42, 3eqtr4di 2793 . . 3 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
5 fveq2 6907 . . . . . . . 8 (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
6 initoval.h . . . . . . . 8 𝐻 = (Hom ‘𝐶)
75, 6eqtr4di 2793 . . . . . . 7 (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
87oveqd 7448 . . . . . 6 (𝑐 = 𝐶 → (𝑏(Hom ‘𝑐)𝑎) = (𝑏𝐻𝑎))
98eleq2d 2825 . . . . 5 (𝑐 = 𝐶 → ( ∈ (𝑏(Hom ‘𝑐)𝑎) ↔ ∈ (𝑏𝐻𝑎)))
109eubidv 2584 . . . 4 (𝑐 = 𝐶 → (∃! ∈ (𝑏(Hom ‘𝑐)𝑎) ↔ ∃! ∈ (𝑏𝐻𝑎)))
114, 10raleqbidv 3344 . . 3 (𝑐 = 𝐶 → (∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑏(Hom ‘𝑐)𝑎) ↔ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑎)))
124, 11rabeqbidv 3452 . 2 (𝑐 = 𝐶 → {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑏(Hom ‘𝑐)𝑎)} = {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑎)})
13 initoval.c . 2 (𝜑𝐶 ∈ Cat)
143fvexi 6921 . . . 4 𝐵 ∈ V
1514rabex 5345 . . 3 {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑎)} ∈ V
1615a1i 11 . 2 (𝜑 → {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑎)} ∈ V)
171, 12, 13, 16fvmptd3 7039 1 (𝜑 → (TermO‘𝐶) = {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑎)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  ∃!weu 2566  wral 3059  {crab 3433  Vcvv 3478  cfv 6563  (class class class)co 7431  Basecbs 17245  Hom chom 17309  Catccat 17709  TermOctermo 18036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-termo 18039
This theorem is referenced by:  istermo  18051  istermoi  18054  dfinito2  18057
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