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| Mirrors > Home > MPE Home > Th. List > Mathboxes > 2arymaptfv | Structured version Visualization version GIF version | ||
| Description: The value of the mapping of binary (endo)functions. (Contributed by AV, 21-May-2024.) |
| Ref | Expression |
|---|---|
| 2arymaptf.h | ⊢ 𝐻 = (ℎ ∈ (2-aryF 𝑋) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉}))) |
| Ref | Expression |
|---|---|
| 2arymaptfv | ⊢ (𝐹 ∈ (2-aryF 𝑋) → (𝐻‘𝐹) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝐹‘{〈0, 𝑥〉, 〈1, 𝑦〉}))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fveq1 6905 | . . 3 ⊢ (ℎ = 𝐹 → (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉}) = (𝐹‘{〈0, 𝑥〉, 〈1, 𝑦〉})) | |
| 2 | 1 | mpoeq3dv 7512 | . 2 ⊢ (ℎ = 𝐹 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉})) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝐹‘{〈0, 𝑥〉, 〈1, 𝑦〉}))) |
| 3 | 2arymaptf.h | . 2 ⊢ 𝐻 = (ℎ ∈ (2-aryF 𝑋) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉}))) | |
| 4 | eqid 2737 | . . . 4 ⊢ (0..^2) = (0..^2) | |
| 5 | 4 | naryrcl 48552 | . . 3 ⊢ (ℎ ∈ (2-aryF 𝑋) → (2 ∈ ℕ0 ∧ 𝑋 ∈ V)) |
| 6 | mpoexga 8102 | . . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑋 ∈ V) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉})) ∈ V) | |
| 7 | 6 | anidms 566 | . . 3 ⊢ (𝑋 ∈ V → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉})) ∈ V) |
| 8 | 5, 7 | simpl2im 503 | . 2 ⊢ (ℎ ∈ (2-aryF 𝑋) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉})) ∈ V) |
| 9 | 2, 3, 8 | fvmpt3 7020 | 1 ⊢ (𝐹 ∈ (2-aryF 𝑋) → (𝐻‘𝐹) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝐹‘{〈0, 𝑥〉, 〈1, 𝑦〉}))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 {cpr 4628 〈cop 4632 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 0cc0 11155 1c1 11156 2c2 12321 ℕ0cn0 12526 ..^cfzo 13694 -aryF cnaryf 48547 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-naryf 48548 |
| This theorem is referenced by: 2arymaptf1 48574 |
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