![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > istermo | Structured version Visualization version GIF version |
Description: The predicate "is a terminal object" of a category. (Contributed by AV, 3-Apr-2020.) |
Ref | Expression |
---|---|
isinito.b | ⊢ 𝐵 = (Base‘𝐶) |
isinito.h | ⊢ 𝐻 = (Hom ‘𝐶) |
isinito.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
isinito.i | ⊢ (𝜑 → 𝐼 ∈ 𝐵) |
Ref | Expression |
---|---|
istermo | ⊢ (𝜑 → (𝐼 ∈ (TermO‘𝐶) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝐼))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isinito.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
2 | isinito.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
3 | isinito.h | . . . 4 ⊢ 𝐻 = (Hom ‘𝐶) | |
4 | 1, 2, 3 | termoval 17990 | . . 3 ⊢ (𝜑 → (TermO‘𝐶) = {𝑖 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑖)}) |
5 | 4 | eleq2d 2815 | . 2 ⊢ (𝜑 → (𝐼 ∈ (TermO‘𝐶) ↔ 𝐼 ∈ {𝑖 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑖)})) |
6 | isinito.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝐵) | |
7 | oveq2 7434 | . . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝑏𝐻𝑖) = (𝑏𝐻𝐼)) | |
8 | 7 | eleq2d 2815 | . . . . . 6 ⊢ (𝑖 = 𝐼 → (ℎ ∈ (𝑏𝐻𝑖) ↔ ℎ ∈ (𝑏𝐻𝐼))) |
9 | 8 | eubidv 2575 | . . . . 5 ⊢ (𝑖 = 𝐼 → (∃!ℎ ℎ ∈ (𝑏𝐻𝑖) ↔ ∃!ℎ ℎ ∈ (𝑏𝐻𝐼))) |
10 | 9 | ralbidv 3175 | . . . 4 ⊢ (𝑖 = 𝐼 → (∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑖) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝐼))) |
11 | 10 | elrab3 3685 | . . 3 ⊢ (𝐼 ∈ 𝐵 → (𝐼 ∈ {𝑖 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑖)} ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝐼))) |
12 | 6, 11 | syl 17 | . 2 ⊢ (𝜑 → (𝐼 ∈ {𝑖 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑖)} ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝐼))) |
13 | 5, 12 | bitrd 278 | 1 ⊢ (𝜑 → (𝐼 ∈ (TermO‘𝐶) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (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 |
Copyright terms: Public domain | W3C validator |