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Theorem termoval 17116
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 17110 . 2 TermO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑏(Hom ‘𝑐)𝑎)})
2 fveq2 6499 . . . 4 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
3 initoval.b . . . 4 𝐵 = (Base‘𝐶)
42, 3syl6eqr 2833 . . 3 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
5 fveq2 6499 . . . . . . . 8 (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
6 initoval.h . . . . . . . 8 𝐻 = (Hom ‘𝐶)
75, 6syl6eqr 2833 . . . . . . 7 (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
87oveqd 6993 . . . . . 6 (𝑐 = 𝐶 → (𝑏(Hom ‘𝑐)𝑎) = (𝑏𝐻𝑎))
98eleq2d 2852 . . . . 5 (𝑐 = 𝐶 → ( ∈ (𝑏(Hom ‘𝑐)𝑎) ↔ ∈ (𝑏𝐻𝑎)))
109eubidv 2605 . . . 4 (𝑐 = 𝐶 → (∃! ∈ (𝑏(Hom ‘𝑐)𝑎) ↔ ∃! ∈ (𝑏𝐻𝑎)))
114, 10raleqbidv 3342 . . 3 (𝑐 = 𝐶 → (∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑏(Hom ‘𝑐)𝑎) ↔ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑎)))
124, 11rabeqbidv 3409 . 2 (𝑐 = 𝐶 → {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑏(Hom ‘𝑐)𝑎)} = {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑎)})
13 initoval.c . 2 (𝜑𝐶 ∈ Cat)
143fvexi 6513 . . . 4 𝐵 ∈ V
1514rabex 5091 . . 3 {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑎)} ∈ V
1615a1i 11 . 2 (𝜑 → {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑎)} ∈ V)
171, 12, 13, 16fvmptd3 6617 1 (𝜑 → (TermO‘𝐶) = {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑎)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1507  wcel 2050  ∃!weu 2583  wral 3089  {crab 3093  Vcvv 3416  cfv 6188  (class class class)co 6976  Basecbs 16339  Hom chom 16432  Catccat 16793  TermOctermo 17107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-sbc 3683  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-iota 6152  df-fun 6190  df-fv 6196  df-ov 6979  df-termo 17110
This theorem is referenced by:  istermo  17119  istermoi  17122
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