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Mirrors > Home > MPE Home > Th. List > termoval | Structured version Visualization version GIF version |
Description: The value of the terminal object function, i.e. the set of all terminal objects of a category. (Contributed by AV, 3-Apr-2020.) |
Ref | Expression |
---|---|
initoval.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
initoval.b | ⊢ 𝐵 = (Base‘𝐶) |
initoval.h | ⊢ 𝐻 = (Hom ‘𝐶) |
Ref | Expression |
---|---|
termoval | ⊢ (𝜑 → (TermO‘𝐶) = {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑎)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-termo 17110 | . 2 ⊢ TermO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝑐)𝑎)}) | |
2 | fveq2 6499 | . . . 4 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶)) | |
3 | initoval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
4 | 2, 3 | syl6eqr 2833 | . . 3 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵) |
5 | fveq2 6499 | . . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶)) | |
6 | initoval.h | . . . . . . . 8 ⊢ 𝐻 = (Hom ‘𝐶) | |
7 | 5, 6 | syl6eqr 2833 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻) |
8 | 7 | oveqd 6993 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑏(Hom ‘𝑐)𝑎) = (𝑏𝐻𝑎)) |
9 | 8 | eleq2d 2852 | . . . . 5 ⊢ (𝑐 = 𝐶 → (ℎ ∈ (𝑏(Hom ‘𝑐)𝑎) ↔ ℎ ∈ (𝑏𝐻𝑎))) |
10 | 9 | eubidv 2605 | . . . 4 ⊢ (𝑐 = 𝐶 → (∃!ℎ ℎ ∈ (𝑏(Hom ‘𝑐)𝑎) ↔ ∃!ℎ ℎ ∈ (𝑏𝐻𝑎))) |
11 | 4, 10 | raleqbidv 3342 | . . 3 ⊢ (𝑐 = 𝐶 → (∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝑐)𝑎) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑎))) |
12 | 4, 11 | rabeqbidv 3409 | . 2 ⊢ (𝑐 = 𝐶 → {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝑐)𝑎)} = {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑎)}) |
13 | initoval.c | . 2 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
14 | 3 | fvexi 6513 | . . . 4 ⊢ 𝐵 ∈ V |
15 | 14 | rabex 5091 | . . 3 ⊢ {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑎)} ∈ V |
16 | 15 | a1i 11 | . 2 ⊢ (𝜑 → {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑎)} ∈ V) |
17 | 1, 12, 13, 16 | fvmptd3 6617 | 1 ⊢ (𝜑 → (TermO‘𝐶) = {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑎)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 ∃!weu 2583 ∀wral 3089 {crab 3093 Vcvv 3416 ‘cfv 6188 (class class class)co 6976 Basecbs 16339 Hom chom 16432 Catccat 16793 TermOctermo 17107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-sep 5060 ax-nul 5067 ax-pr 5186 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3418 df-sbc 3683 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-nul 4180 df-if 4351 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-iota 6152 df-fun 6190 df-fv 6196 df-ov 6979 df-termo 17110 |
This theorem is referenced by: istermo 17119 istermoi 17122 |
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