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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 768 | . . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ ℎ ∈ (2-aryF 𝑋)) ∧ 𝑧 ∈ (𝑋 × 𝑋)) → ℎ ∈ (2-aryF 𝑋)) | |
| 2 | xp1st 8047 | . . . . . 6 ⊢ (𝑧 ∈ (𝑋 × 𝑋) → (1st ‘𝑧) ∈ 𝑋) | |
| 3 | 2 | adantl 481 | . . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ ℎ ∈ (2-aryF 𝑋)) ∧ 𝑧 ∈ (𝑋 × 𝑋)) → (1st ‘𝑧) ∈ 𝑋) | 
| 4 | xp2nd 8048 | . . . . . 6 ⊢ (𝑧 ∈ (𝑋 × 𝑋) → (2nd ‘𝑧) ∈ 𝑋) | |
| 5 | 4 | adantl 481 | . . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ ℎ ∈ (2-aryF 𝑋)) ∧ 𝑧 ∈ (𝑋 × 𝑋)) → (2nd ‘𝑧) ∈ 𝑋) | 
| 6 | fv2arycl 48574 | . . . . 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 1372 | . . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ ℎ ∈ (2-aryF 𝑋)) ∧ 𝑧 ∈ (𝑋 × 𝑋)) → (ℎ‘{〈0, (1st ‘𝑧)〉, 〈1, (2nd ‘𝑧)〉}) ∈ 𝑋) | 
| 8 | vex 3483 | . . . . . . . . . 10 ⊢ 𝑥 ∈ V | |
| 9 | vex 3483 | . . . . . . . . . 10 ⊢ 𝑦 ∈ V | |
| 10 | 8, 9 | op1std 8025 | . . . . . . . . 9 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (1st ‘𝑧) = 𝑥) | 
| 11 | 10 | opeq2d 4879 | . . . . . . . 8 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → 〈0, (1st ‘𝑧)〉 = 〈0, 𝑥〉) | 
| 12 | 8, 9 | op2ndd 8026 | . . . . . . . . 9 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (2nd ‘𝑧) = 𝑦) | 
| 13 | 12 | opeq2d 4879 | . . . . . . . 8 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → 〈1, (2nd ‘𝑧)〉 = 〈1, 𝑦〉) | 
| 14 | 11, 13 | preq12d 4740 | . . . . . . 7 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → {〈0, (1st ‘𝑧)〉, 〈1, (2nd ‘𝑧)〉} = {〈0, 𝑥〉, 〈1, 𝑦〉}) | 
| 15 | 14 | fveq2d 6909 | . . . . . 6 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (ℎ‘{〈0, (1st ‘𝑧)〉, 〈1, (2nd ‘𝑧)〉}) = (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉})) | 
| 16 | 15 | mpompt 7548 | . . . . 5 ⊢ (𝑧 ∈ (𝑋 × 𝑋) ↦ (ℎ‘{〈0, (1st ‘𝑧)〉, 〈1, (2nd ‘𝑧)〉})) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉})) | 
| 17 | 16 | eqcomi 2745 | . . . 4 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉})) = (𝑧 ∈ (𝑋 × 𝑋) ↦ (ℎ‘{〈0, (1st ‘𝑧)〉, 〈1, (2nd ‘𝑧)〉})) | 
| 18 | 7, 17 | fmptd 7133 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ℎ ∈ (2-aryF 𝑋)) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉})):(𝑋 × 𝑋)⟶𝑋) | 
| 19 | sqxpexg 7776 | . . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝑋 × 𝑋) ∈ V) | |
| 20 | elmapg 8880 | . . . . 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 480 | . . 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 257 | . 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 7133 | 1 ⊢ (𝑋 ∈ 𝑉 → 𝐻:(2-aryF 𝑋)⟶(𝑋 ↑m (𝑋 × 𝑋))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 Vcvv 3479 {cpr 4627 〈cop 4631 ↦ cmpt 5224 × cxp 5682 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ∈ cmpo 7434 1st c1st 8013 2nd c2nd 8014 ↑m cmap 8867 0cc0 11156 1c1 11157 2c2 12322 -aryF cnaryf 48552 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-n0 12529 df-z 12616 df-uz 12880 df-fz 13549 df-fzo 13696 df-naryf 48553 | 
| This theorem is referenced by: 2arymaptf1 48579 2arymaptfo 48580 | 
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