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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 16648 | . . 3 ⊢ (𝜑 → (TermO‘𝐶) = {𝑖 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑖)}) |
5 | 4 | eleq2d 2687 | . 2 ⊢ (𝜑 → (𝐼 ∈ (TermO‘𝐶) ↔ 𝐼 ∈ {𝑖 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑖)})) |
6 | isinito.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝐵) | |
7 | oveq2 6658 | . . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝑏𝐻𝑖) = (𝑏𝐻𝐼)) | |
8 | 7 | eleq2d 2687 | . . . . . 6 ⊢ (𝑖 = 𝐼 → (ℎ ∈ (𝑏𝐻𝑖) ↔ ℎ ∈ (𝑏𝐻𝐼))) |
9 | 8 | eubidv 2490 | . . . . 5 ⊢ (𝑖 = 𝐼 → (∃!ℎ ℎ ∈ (𝑏𝐻𝑖) ↔ ∃!ℎ ℎ ∈ (𝑏𝐻𝐼))) |
10 | 9 | ralbidv 2986 | . . . 4 ⊢ (𝑖 = 𝐼 → (∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑖) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝐼))) |
11 | 10 | elrab3 3364 | . . 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 268 | 1 ⊢ (𝜑 → (𝐼 ∈ (TermO‘𝐶) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝐼))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 ∈ wcel 1990 ∃!weu 2470 ∀wral 2912 {crab 2916 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 Hom chom 15952 Catccat 16325 TermOctermo 16639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-termo 16642 |
This theorem is referenced by: istermoi 16654 zrtermorngc 42001 zrtermoringc 42070 |
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