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Theorem istermo 17010
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 17007 . . 3 (𝜑 → (TermO‘𝐶) = {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑖)})
54eleq2d 2892 . 2 (𝜑 → (𝐼 ∈ (TermO‘𝐶) ↔ 𝐼 ∈ {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑖)}))
6 isinito.i . . 3 (𝜑𝐼𝐵)
7 oveq2 6918 . . . . . . 7 (𝑖 = 𝐼 → (𝑏𝐻𝑖) = (𝑏𝐻𝐼))
87eleq2d 2892 . . . . . 6 (𝑖 = 𝐼 → ( ∈ (𝑏𝐻𝑖) ↔ ∈ (𝑏𝐻𝐼)))
98eubidv 2659 . . . . 5 (𝑖 = 𝐼 → (∃! ∈ (𝑏𝐻𝑖) ↔ ∃! ∈ (𝑏𝐻𝐼)))
109ralbidv 3195 . . . 4 (𝑖 = 𝐼 → (∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑖) ↔ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝐼)))
1110elrab3 3587 . . 3 (𝐼𝐵 → (𝐼 ∈ {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑖)} ↔ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝐼)))
126, 11syl 17 . 2 (𝜑 → (𝐼 ∈ {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑖)} ↔ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝐼)))
135, 12bitrd 271 1 (𝜑 → (𝐼 ∈ (TermO‘𝐶) ↔ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝐼)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1656  wcel 2164  ∃!weu 2639  wral 3117  {crab 3121  cfv 6127  (class class class)co 6910  Basecbs 16229  Hom chom 16323  Catccat 16684  TermOctermo 16998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-iota 6090  df-fun 6129  df-fv 6135  df-ov 6913  df-termo 17001
This theorem is referenced by:  istermoi  17013  zrtermorngc  42862  zrtermoringc  42931
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