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Theorem istermo 17993
Description: The predicate "is a terminal object" of a category. (Contributed by AV, 3-Apr-2020.)
Hypotheses
Ref Expression
isinito.b 𝐵 = (Base‘𝐶)
isinito.h 𝐻 = (Hom ‘𝐶)
isinito.c (𝜑𝐶 ∈ Cat)
isinito.i (𝜑𝐼𝐵)
Assertion
Ref Expression
istermo (𝜑 → (𝐼 ∈ (TermO‘𝐶) ↔ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝐼)))
Distinct variable groups:   𝐵,𝑏   𝐶,𝑏,   𝐼,𝑏,
Allowed substitution hints:   𝜑(,𝑏)   𝐵()   𝐻(,𝑏)

Proof of Theorem istermo
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 isinito.c . . . 4 (𝜑𝐶 ∈ Cat)
2 isinito.b . . . 4 𝐵 = (Base‘𝐶)
3 isinito.h . . . 4 𝐻 = (Hom ‘𝐶)
41, 2, 3termoval 17990 . . 3 (𝜑 → (TermO‘𝐶) = {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑖)})
54eleq2d 2815 . 2 (𝜑 → (𝐼 ∈ (TermO‘𝐶) ↔ 𝐼 ∈ {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑖)}))
6 isinito.i . . 3 (𝜑𝐼𝐵)
7 oveq2 7434 . . . . . . 7 (𝑖 = 𝐼 → (𝑏𝐻𝑖) = (𝑏𝐻𝐼))
87eleq2d 2815 . . . . . 6 (𝑖 = 𝐼 → ( ∈ (𝑏𝐻𝑖) ↔ ∈ (𝑏𝐻𝐼)))
98eubidv 2575 . . . . 5 (𝑖 = 𝐼 → (∃! ∈ (𝑏𝐻𝑖) ↔ ∃! ∈ (𝑏𝐻𝐼)))
109ralbidv 3175 . . . 4 (𝑖 = 𝐼 → (∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑖) ↔ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝐼)))
1110elrab3 3685 . . 3 (𝐼𝐵 → (𝐼 ∈ {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑖)} ↔ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝐼)))
126, 11syl 17 . 2 (𝜑 → (𝐼 ∈ {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑖)} ↔ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝐼)))
135, 12bitrd 278 1 (𝜑 → (𝐼 ∈ (TermO‘𝐶) ↔ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝐼)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wcel 2098  ∃!weu 2557  wral 3058  {crab 3430  cfv 6553  (class class class)co 7426  Basecbs 17187  Hom chom 17251  Catccat 17651  TermOctermo 17978
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 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-termo 17981
This theorem is referenced by:  istermoi  17996  zrtermorngc  20583  zrtermoringc  20615
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