Mathbox for Alexander van der Vekens |
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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 6675 | . . 3 ⊢ (ℎ = 𝐹 → (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉}) = (𝐹‘{〈0, 𝑥〉, 〈1, 𝑦〉})) | |
2 | 1 | mpoeq3dv 7249 | . 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 2738 | . . . 4 ⊢ (0..^2) = (0..^2) | |
5 | 4 | naryrcl 45540 | . . 3 ⊢ (ℎ ∈ (2-aryF 𝑋) → (2 ∈ ℕ0 ∧ 𝑋 ∈ V)) |
6 | mpoexga 7803 | . . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑋 ∈ V) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉})) ∈ V) | |
7 | 6 | anidms 570 | . . 3 ⊢ (𝑋 ∈ V → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉})) ∈ V) |
8 | 5, 7 | simpl2im 507 | . 2 ⊢ (ℎ ∈ (2-aryF 𝑋) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉})) ∈ V) |
9 | 2, 3, 8 | fvmpt3 6781 | 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 1542 ∈ wcel 2114 Vcvv 3398 {cpr 4518 〈cop 4522 ↦ cmpt 5110 ‘cfv 6339 (class class class)co 7172 ∈ cmpo 7174 0cc0 10617 1c1 10618 2c2 11773 ℕ0cn0 11978 ..^cfzo 13126 -aryF cnaryf 45535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-ov 7175 df-oprab 7176 df-mpo 7177 df-1st 7716 df-2nd 7717 df-naryf 45536 |
This theorem is referenced by: 2arymaptf1 45562 |
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