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Mirrors > Home > MPE Home > Th. List > initoval | Structured version Visualization version GIF version |
Description: The value of the initial object function, i.e. the set of all initial 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 |
---|---|
initoval | ⊢ (𝜑 → (InitO‘𝐶) = {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑎𝐻𝑏)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inito 17830 | . 2 ⊢ InitO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏)}) | |
2 | fveq2 6840 | . . . 4 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶)) | |
3 | initoval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
4 | 2, 3 | eqtr4di 2796 | . . 3 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵) |
5 | fveq2 6840 | . . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶)) | |
6 | initoval.h | . . . . . . . 8 ⊢ 𝐻 = (Hom ‘𝐶) | |
7 | 5, 6 | eqtr4di 2796 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻) |
8 | 7 | oveqd 7369 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑎(Hom ‘𝑐)𝑏) = (𝑎𝐻𝑏)) |
9 | 8 | eleq2d 2824 | . . . . 5 ⊢ (𝑐 = 𝐶 → (ℎ ∈ (𝑎(Hom ‘𝑐)𝑏) ↔ ℎ ∈ (𝑎𝐻𝑏))) |
10 | 9 | eubidv 2586 | . . . 4 ⊢ (𝑐 = 𝐶 → (∃!ℎ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏) ↔ ∃!ℎ ℎ ∈ (𝑎𝐻𝑏))) |
11 | 4, 10 | raleqbidv 3318 | . . 3 ⊢ (𝑐 = 𝐶 → (∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑎𝐻𝑏))) |
12 | 4, 11 | rabeqbidv 3423 | . 2 ⊢ (𝑐 = 𝐶 → {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏)} = {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑎𝐻𝑏)}) |
13 | initoval.c | . 2 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
14 | 3 | fvexi 6854 | . . . 4 ⊢ 𝐵 ∈ V |
15 | 14 | rabex 5288 | . . 3 ⊢ {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑎𝐻𝑏)} ∈ V |
16 | 15 | a1i 11 | . 2 ⊢ (𝜑 → {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑎𝐻𝑏)} ∈ V) |
17 | 1, 12, 13, 16 | fvmptd3 6969 | 1 ⊢ (𝜑 → (InitO‘𝐶) = {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑎𝐻𝑏)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∃!weu 2568 ∀wral 3063 {crab 3406 Vcvv 3444 ‘cfv 6494 (class class class)co 7352 Basecbs 17043 Hom chom 17104 Catccat 17504 InitOcinito 17827 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-iota 6446 df-fun 6496 df-fv 6502 df-ov 7355 df-inito 17830 |
This theorem is referenced by: isinito 17842 isinitoi 17845 dftermo2 17850 |
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