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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 6650 | . . 3 ⊢ (ℎ = 𝐹 → (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉}) = (𝐹‘{〈0, 𝑥〉, 〈1, 𝑦〉})) | |
2 | 1 | mpoeq3dv 7220 | . 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 2759 | . . . 4 ⊢ (0..^2) = (0..^2) | |
5 | 4 | naryrcl 45395 | . . 3 ⊢ (ℎ ∈ (2-aryF 𝑋) → (2 ∈ ℕ0 ∧ 𝑋 ∈ V)) |
6 | mpoexga 7773 | . . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑋 ∈ V) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉})) ∈ V) | |
7 | 6 | anidms 571 | . . 3 ⊢ (𝑋 ∈ V → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉})) ∈ V) |
8 | 5, 7 | simpl2im 508 | . 2 ⊢ (ℎ ∈ (2-aryF 𝑋) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉})) ∈ V) |
9 | 2, 3, 8 | fvmpt3 6756 | 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 1539 ∈ wcel 2112 Vcvv 3407 {cpr 4517 〈cop 4521 ↦ cmpt 5105 ‘cfv 6328 (class class class)co 7143 ∈ cmpo 7145 0cc0 10560 1c1 10561 2c2 11714 ℕ0cn0 11919 ..^cfzo 13067 -aryF cnaryf 45390 |
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 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-ral 3073 df-rex 3074 df-reu 3075 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-op 4522 df-uni 4792 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-id 5423 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-ov 7146 df-oprab 7147 df-mpo 7148 df-1st 7686 df-2nd 7687 df-naryf 45391 |
This theorem is referenced by: 2arymaptf1 45417 |
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