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Theorem termoval 17986
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 17977 . 2 TermO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑏(Hom ‘𝑐)𝑎)})
2 fveq2 6896 . . . 4 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
3 initoval.b . . . 4 𝐵 = (Base‘𝐶)
42, 3eqtr4di 2783 . . 3 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
5 fveq2 6896 . . . . . . . 8 (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
6 initoval.h . . . . . . . 8 𝐻 = (Hom ‘𝐶)
75, 6eqtr4di 2783 . . . . . . 7 (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
87oveqd 7436 . . . . . 6 (𝑐 = 𝐶 → (𝑏(Hom ‘𝑐)𝑎) = (𝑏𝐻𝑎))
98eleq2d 2811 . . . . 5 (𝑐 = 𝐶 → ( ∈ (𝑏(Hom ‘𝑐)𝑎) ↔ ∈ (𝑏𝐻𝑎)))
109eubidv 2574 . . . 4 (𝑐 = 𝐶 → (∃! ∈ (𝑏(Hom ‘𝑐)𝑎) ↔ ∃! ∈ (𝑏𝐻𝑎)))
114, 10raleqbidv 3329 . . 3 (𝑐 = 𝐶 → (∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑏(Hom ‘𝑐)𝑎) ↔ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑎)))
124, 11rabeqbidv 3436 . 2 (𝑐 = 𝐶 → {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑏(Hom ‘𝑐)𝑎)} = {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑎)})
13 initoval.c . 2 (𝜑𝐶 ∈ Cat)
143fvexi 6910 . . . 4 𝐵 ∈ V
1514rabex 5335 . . 3 {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑎)} ∈ V
1615a1i 11 . 2 (𝜑 → {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑎)} ∈ V)
171, 12, 13, 16fvmptd3 7027 1 (𝜑 → (TermO‘𝐶) = {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑎)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  ∃!weu 2556  wral 3050  {crab 3418  Vcvv 3461  cfv 6549  (class class class)co 7419  Basecbs 17183  Hom chom 17247  Catccat 17647  TermOctermo 17974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fv 6557  df-ov 7422  df-termo 17977
This theorem is referenced by:  istermo  17989  istermoi  17992  dfinito2  17995
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