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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 17977 | . 2 ⊢ TermO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝑐)𝑎)}) | |
2 | fveq2 6896 | . . . 4 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶)) | |
3 | initoval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
4 | 2, 3 | eqtr4di 2783 | . . 3 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵) |
5 | fveq2 6896 | . . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶)) | |
6 | initoval.h | . . . . . . . 8 ⊢ 𝐻 = (Hom ‘𝐶) | |
7 | 5, 6 | eqtr4di 2783 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻) |
8 | 7 | oveqd 7436 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑏(Hom ‘𝑐)𝑎) = (𝑏𝐻𝑎)) |
9 | 8 | eleq2d 2811 | . . . . 5 ⊢ (𝑐 = 𝐶 → (ℎ ∈ (𝑏(Hom ‘𝑐)𝑎) ↔ ℎ ∈ (𝑏𝐻𝑎))) |
10 | 9 | eubidv 2574 | . . . 4 ⊢ (𝑐 = 𝐶 → (∃!ℎ ℎ ∈ (𝑏(Hom ‘𝑐)𝑎) ↔ ∃!ℎ ℎ ∈ (𝑏𝐻𝑎))) |
11 | 4, 10 | raleqbidv 3329 | . . 3 ⊢ (𝑐 = 𝐶 → (∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝑐)𝑎) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑎))) |
12 | 4, 11 | rabeqbidv 3436 | . 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 6910 | . . . 4 ⊢ 𝐵 ∈ V |
15 | 14 | rabex 5335 | . . 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 7027 | 1 ⊢ (𝜑 → (TermO‘𝐶) = {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑎)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∃!weu 2556 ∀wral 3050 {crab 3418 Vcvv 3461 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 Hom chom 17247 Catccat 17647 TermOctermo 17974 |
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 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6501 df-fun 6551 df-fv 6557 df-ov 7422 df-termo 17977 |
This theorem is referenced by: istermo 17989 istermoi 17992 dfinito2 17995 |
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