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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2arymaptf | Structured version Visualization version GIF version |
Description: The mapping of binary (endo)functions is a function into the set of binary operations. (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 |
---|---|
2arymaptf | ⊢ (𝑋 ∈ 𝑉 → 𝐻:(2-aryF 𝑋)⟶(𝑋 ↑m (𝑋 × 𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplr 769 | . . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ ℎ ∈ (2-aryF 𝑋)) ∧ 𝑧 ∈ (𝑋 × 𝑋)) → ℎ ∈ (2-aryF 𝑋)) | |
2 | xp1st 7718 | . . . . . 6 ⊢ (𝑧 ∈ (𝑋 × 𝑋) → (1st ‘𝑧) ∈ 𝑋) | |
3 | 2 | adantl 486 | . . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ ℎ ∈ (2-aryF 𝑋)) ∧ 𝑧 ∈ (𝑋 × 𝑋)) → (1st ‘𝑧) ∈ 𝑋) |
4 | xp2nd 7719 | . . . . . 6 ⊢ (𝑧 ∈ (𝑋 × 𝑋) → (2nd ‘𝑧) ∈ 𝑋) | |
5 | 4 | adantl 486 | . . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ ℎ ∈ (2-aryF 𝑋)) ∧ 𝑧 ∈ (𝑋 × 𝑋)) → (2nd ‘𝑧) ∈ 𝑋) |
6 | fv2arycl 45412 | . . . . 5 ⊢ ((ℎ ∈ (2-aryF 𝑋) ∧ (1st ‘𝑧) ∈ 𝑋 ∧ (2nd ‘𝑧) ∈ 𝑋) → (ℎ‘{〈0, (1st ‘𝑧)〉, 〈1, (2nd ‘𝑧)〉}) ∈ 𝑋) | |
7 | 1, 3, 5, 6 | syl3anc 1369 | . . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ ℎ ∈ (2-aryF 𝑋)) ∧ 𝑧 ∈ (𝑋 × 𝑋)) → (ℎ‘{〈0, (1st ‘𝑧)〉, 〈1, (2nd ‘𝑧)〉}) ∈ 𝑋) |
8 | vex 3411 | . . . . . . . . . 10 ⊢ 𝑥 ∈ V | |
9 | vex 3411 | . . . . . . . . . 10 ⊢ 𝑦 ∈ V | |
10 | 8, 9 | op1std 7696 | . . . . . . . . 9 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (1st ‘𝑧) = 𝑥) |
11 | 10 | opeq2d 4763 | . . . . . . . 8 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → 〈0, (1st ‘𝑧)〉 = 〈0, 𝑥〉) |
12 | 8, 9 | op2ndd 7697 | . . . . . . . . 9 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (2nd ‘𝑧) = 𝑦) |
13 | 12 | opeq2d 4763 | . . . . . . . 8 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → 〈1, (2nd ‘𝑧)〉 = 〈1, 𝑦〉) |
14 | 11, 13 | preq12d 4627 | . . . . . . 7 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → {〈0, (1st ‘𝑧)〉, 〈1, (2nd ‘𝑧)〉} = {〈0, 𝑥〉, 〈1, 𝑦〉}) |
15 | 14 | fveq2d 6655 | . . . . . 6 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (ℎ‘{〈0, (1st ‘𝑧)〉, 〈1, (2nd ‘𝑧)〉}) = (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉})) |
16 | 15 | mpompt 7253 | . . . . 5 ⊢ (𝑧 ∈ (𝑋 × 𝑋) ↦ (ℎ‘{〈0, (1st ‘𝑧)〉, 〈1, (2nd ‘𝑧)〉})) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉})) |
17 | 16 | eqcomi 2768 | . . . 4 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉})) = (𝑧 ∈ (𝑋 × 𝑋) ↦ (ℎ‘{〈0, (1st ‘𝑧)〉, 〈1, (2nd ‘𝑧)〉})) |
18 | 7, 17 | fmptd 6862 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ℎ ∈ (2-aryF 𝑋)) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉})):(𝑋 × 𝑋)⟶𝑋) |
19 | sqxpexg 7469 | . . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝑋 × 𝑋) ∈ V) | |
20 | elmapg 8422 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑋 × 𝑋) ∈ V) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉})) ∈ (𝑋 ↑m (𝑋 × 𝑋)) ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉})):(𝑋 × 𝑋)⟶𝑋)) | |
21 | 19, 20 | mpdan 687 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉})) ∈ (𝑋 ↑m (𝑋 × 𝑋)) ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉})):(𝑋 × 𝑋)⟶𝑋)) |
22 | 21 | adantr 485 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ℎ ∈ (2-aryF 𝑋)) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉})) ∈ (𝑋 ↑m (𝑋 × 𝑋)) ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉})):(𝑋 × 𝑋)⟶𝑋)) |
23 | 18, 22 | mpbird 260 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ℎ ∈ (2-aryF 𝑋)) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉})) ∈ (𝑋 ↑m (𝑋 × 𝑋))) |
24 | 2arymaptf.h | . 2 ⊢ 𝐻 = (ℎ ∈ (2-aryF 𝑋) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉}))) | |
25 | 23, 24 | fmptd 6862 | 1 ⊢ (𝑋 ∈ 𝑉 → 𝐻:(2-aryF 𝑋)⟶(𝑋 ↑m (𝑋 × 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1539 ∈ wcel 2112 Vcvv 3407 {cpr 4517 〈cop 4521 ↦ cmpt 5105 × cxp 5515 ⟶wf 6324 ‘cfv 6328 (class class class)co 7143 ∈ cmpo 7145 1st c1st 7684 2nd c2nd 7685 ↑m cmap 8409 0cc0 10560 1c1 10561 2c2 11714 -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-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 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-nel 3054 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-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 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-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 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-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-om 7573 df-1st 7686 df-2nd 7687 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-map 8411 df-en 8521 df-dom 8522 df-sdom 8523 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-nn 11660 df-2 11722 df-n0 11920 df-z 12006 df-uz 12268 df-fz 12925 df-fzo 13068 df-naryf 45391 |
This theorem is referenced by: 2arymaptf1 45417 2arymaptfo 45418 |
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