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Mirrors > Home > MPE Home > Th. List > homaf | Structured version Visualization version GIF version |
Description: Functionality of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
homarcl.h | ⊢ 𝐻 = (Homa‘𝐶) |
homafval.b | ⊢ 𝐵 = (Base‘𝐶) |
homafval.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
Ref | Expression |
---|---|
homaf | ⊢ (𝜑 → 𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snssi 4526 | . . . . . 6 ⊢ (𝑥 ∈ (𝐵 × 𝐵) → {𝑥} ⊆ (𝐵 × 𝐵)) | |
2 | 1 | adantl 474 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 × 𝐵)) → {𝑥} ⊆ (𝐵 × 𝐵)) |
3 | ssv 3820 | . . . . 5 ⊢ ((Hom ‘𝐶)‘𝑥) ⊆ V | |
4 | xpss12 5326 | . . . . 5 ⊢ (({𝑥} ⊆ (𝐵 × 𝐵) ∧ ((Hom ‘𝐶)‘𝑥) ⊆ V) → ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ⊆ ((𝐵 × 𝐵) × V)) | |
5 | 2, 3, 4 | sylancl 581 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 × 𝐵)) → ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ⊆ ((𝐵 × 𝐵) × V)) |
6 | snex 5098 | . . . . . 6 ⊢ {𝑥} ∈ V | |
7 | fvex 6423 | . . . . . 6 ⊢ ((Hom ‘𝐶)‘𝑥) ∈ V | |
8 | 6, 7 | xpex 7195 | . . . . 5 ⊢ ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ∈ V |
9 | 8 | elpw 4354 | . . . 4 ⊢ (({𝑥} × ((Hom ‘𝐶)‘𝑥)) ∈ 𝒫 ((𝐵 × 𝐵) × V) ↔ ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ⊆ ((𝐵 × 𝐵) × V)) |
10 | 5, 9 | sylibr 226 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 × 𝐵)) → ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ∈ 𝒫 ((𝐵 × 𝐵) × V)) |
11 | 10 | fmpttd 6610 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × ((Hom ‘𝐶)‘𝑥))):(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V)) |
12 | homarcl.h | . . . 4 ⊢ 𝐻 = (Homa‘𝐶) | |
13 | homafval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
14 | homafval.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
15 | eqid 2798 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
16 | 12, 13, 14, 15 | homafval 16990 | . . 3 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × ((Hom ‘𝐶)‘𝑥)))) |
17 | 16 | feq1d 6240 | . 2 ⊢ (𝜑 → (𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V) ↔ (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × ((Hom ‘𝐶)‘𝑥))):(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V))) |
18 | 11, 17 | mpbird 249 | 1 ⊢ (𝜑 → 𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 Vcvv 3384 ⊆ wss 3768 𝒫 cpw 4348 {csn 4367 ↦ cmpt 4921 × cxp 5309 ⟶wf 6096 ‘cfv 6100 Basecbs 16181 Hom chom 16275 Catccat 16636 Homachoma 16984 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2776 ax-rep 4963 ax-sep 4974 ax-nul 4982 ax-pow 5034 ax-pr 5096 ax-un 7182 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2785 df-cleq 2791 df-clel 2794 df-nfc 2929 df-ne 2971 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3386 df-sbc 3633 df-csb 3728 df-dif 3771 df-un 3773 df-in 3775 df-ss 3782 df-nul 4115 df-if 4277 df-pw 4350 df-sn 4368 df-pr 4370 df-op 4374 df-uni 4628 df-iun 4711 df-br 4843 df-opab 4905 df-mpt 4922 df-id 5219 df-xp 5317 df-rel 5318 df-cnv 5319 df-co 5320 df-dm 5321 df-rn 5322 df-res 5323 df-ima 5324 df-iota 6063 df-fun 6102 df-fn 6103 df-f 6104 df-f1 6105 df-fo 6106 df-f1o 6107 df-fv 6108 df-homa 16987 |
This theorem is referenced by: homarcl2 16996 homarel 16997 arwhoma 17006 |
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