MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  istermo Structured version   Visualization version   GIF version

Theorem istermo 17240
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 17237 . . 3 (𝜑 → (TermO‘𝐶) = {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑖)})
54eleq2d 2896 . 2 (𝜑 → (𝐼 ∈ (TermO‘𝐶) ↔ 𝐼 ∈ {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑖)}))
6 isinito.i . . 3 (𝜑𝐼𝐵)
7 oveq2 7141 . . . . . . 7 (𝑖 = 𝐼 → (𝑏𝐻𝑖) = (𝑏𝐻𝐼))
87eleq2d 2896 . . . . . 6 (𝑖 = 𝐼 → ( ∈ (𝑏𝐻𝑖) ↔ ∈ (𝑏𝐻𝐼)))
98eubidv 2671 . . . . 5 (𝑖 = 𝐼 → (∃! ∈ (𝑏𝐻𝑖) ↔ ∃! ∈ (𝑏𝐻𝐼)))
109ralbidv 3184 . . . 4 (𝑖 = 𝐼 → (∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑖) ↔ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝐼)))
1110elrab3 3661 . . 3 (𝐼𝐵 → (𝐼 ∈ {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑖)} ↔ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝐼)))
126, 11syl 17 . 2 (𝜑 → (𝐼 ∈ {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑖)} ↔ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝐼)))
135, 12bitrd 281 1 (𝜑 → (𝐼 ∈ (TermO‘𝐶) ↔ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝐼)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208   = wceq 1537  wcel 2114  ∃!weu 2652  wral 3125  {crab 3129  cfv 6331  (class class class)co 7133  Basecbs 16462  Hom chom 16555  Catccat 16914  TermOctermo 17228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-iota 6290  df-fun 6333  df-fv 6339  df-ov 7136  df-termo 17231
This theorem is referenced by:  istermoi  17243  zrtermorngc  44417  zrtermoringc  44486
  Copyright terms: Public domain W3C validator