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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 | homarcl.h | . . 3 ⊢ 𝐻 = (Homa‘𝐶) | |
2 | homafval.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
3 | homafval.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
4 | eqid 2732 | . . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
5 | 1, 2, 3, 4 | homafval 17975 | . 2 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × ((Hom ‘𝐶)‘𝑥)))) |
6 | snssi 4810 | . . . . 5 ⊢ (𝑥 ∈ (𝐵 × 𝐵) → {𝑥} ⊆ (𝐵 × 𝐵)) | |
7 | 6 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 × 𝐵)) → {𝑥} ⊆ (𝐵 × 𝐵)) |
8 | ssv 4005 | . . . 4 ⊢ ((Hom ‘𝐶)‘𝑥) ⊆ V | |
9 | xpss12 5690 | . . . 4 ⊢ (({𝑥} ⊆ (𝐵 × 𝐵) ∧ ((Hom ‘𝐶)‘𝑥) ⊆ V) → ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ⊆ ((𝐵 × 𝐵) × V)) | |
10 | 7, 8, 9 | sylancl 586 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 × 𝐵)) → ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ⊆ ((𝐵 × 𝐵) × V)) |
11 | vsnex 5428 | . . . . 5 ⊢ {𝑥} ∈ V | |
12 | fvex 6901 | . . . . 5 ⊢ ((Hom ‘𝐶)‘𝑥) ∈ V | |
13 | 11, 12 | xpex 7736 | . . . 4 ⊢ ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ∈ V |
14 | 13 | elpw 4605 | . . 3 ⊢ (({𝑥} × ((Hom ‘𝐶)‘𝑥)) ∈ 𝒫 ((𝐵 × 𝐵) × V) ↔ ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ⊆ ((𝐵 × 𝐵) × V)) |
15 | 10, 14 | sylibr 233 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 × 𝐵)) → ({𝑥} × ((Hom ‘𝐶)‘𝑥)) ∈ 𝒫 ((𝐵 × 𝐵) × V)) |
16 | 5, 15 | fmpt3d 7112 | 1 ⊢ (𝜑 → 𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ⊆ wss 3947 𝒫 cpw 4601 {csn 4627 × cxp 5673 ⟶wf 6536 ‘cfv 6540 Basecbs 17140 Hom chom 17204 Catccat 17604 Homachoma 17969 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-homa 17972 |
This theorem is referenced by: homarcl2 17981 homarel 17982 arwhoma 17991 |
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