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Theorem istermo 16858
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 16855 . . 3 (𝜑 → (TermO‘𝐶) = {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑖)})
54eleq2d 2836 . 2 (𝜑 → (𝐼 ∈ (TermO‘𝐶) ↔ 𝐼 ∈ {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑖)}))
6 isinito.i . . 3 (𝜑𝐼𝐵)
7 oveq2 6804 . . . . . . 7 (𝑖 = 𝐼 → (𝑏𝐻𝑖) = (𝑏𝐻𝐼))
87eleq2d 2836 . . . . . 6 (𝑖 = 𝐼 → ( ∈ (𝑏𝐻𝑖) ↔ ∈ (𝑏𝐻𝐼)))
98eubidv 2638 . . . . 5 (𝑖 = 𝐼 → (∃! ∈ (𝑏𝐻𝑖) ↔ ∃! ∈ (𝑏𝐻𝐼)))
109ralbidv 3135 . . . 4 (𝑖 = 𝐼 → (∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑖) ↔ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝐼)))
1110elrab3 3516 . . 3 (𝐼𝐵 → (𝐼 ∈ {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑖)} ↔ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝐼)))
126, 11syl 17 . 2 (𝜑 → (𝐼 ∈ {𝑖𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑖)} ↔ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝐼)))
135, 12bitrd 268 1 (𝜑 → (𝐼 ∈ (TermO‘𝐶) ↔ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝐼)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  wcel 2145  ∃!weu 2618  wral 3061  {crab 3065  cfv 6030  (class class class)co 6796  Basecbs 16064  Hom chom 16160  Catccat 16532  TermOctermo 16846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fv 6038  df-ov 6799  df-termo 16849
This theorem is referenced by:  istermoi  16861  zrtermorngc  42524  zrtermoringc  42593
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