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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 8001 | . . . . . 6 ⊢ (𝑧 ∈ (𝑋 × 𝑋) → (1st ‘𝑧) ∈ 𝑋) | |
3 | 2 | adantl 483 | . . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ ℎ ∈ (2-aryF 𝑋)) ∧ 𝑧 ∈ (𝑋 × 𝑋)) → (1st ‘𝑧) ∈ 𝑋) |
4 | xp2nd 8002 | . . . . . 6 ⊢ (𝑧 ∈ (𝑋 × 𝑋) → (2nd ‘𝑧) ∈ 𝑋) | |
5 | 4 | adantl 483 | . . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ ℎ ∈ (2-aryF 𝑋)) ∧ 𝑧 ∈ (𝑋 × 𝑋)) → (2nd ‘𝑧) ∈ 𝑋) |
6 | fv2arycl 47235 | . . . . 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 3479 | . . . . . . . . . 10 ⊢ 𝑥 ∈ V | |
9 | vex 3479 | . . . . . . . . . 10 ⊢ 𝑦 ∈ V | |
10 | 8, 9 | op1std 7979 | . . . . . . . . 9 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (1st ‘𝑧) = 𝑥) |
11 | 10 | opeq2d 4878 | . . . . . . . 8 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → 〈0, (1st ‘𝑧)〉 = 〈0, 𝑥〉) |
12 | 8, 9 | op2ndd 7980 | . . . . . . . . 9 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (2nd ‘𝑧) = 𝑦) |
13 | 12 | opeq2d 4878 | . . . . . . . 8 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → 〈1, (2nd ‘𝑧)〉 = 〈1, 𝑦〉) |
14 | 11, 13 | preq12d 4743 | . . . . . . 7 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → {〈0, (1st ‘𝑧)〉, 〈1, (2nd ‘𝑧)〉} = {〈0, 𝑥〉, 〈1, 𝑦〉}) |
15 | 14 | fveq2d 6891 | . . . . . 6 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (ℎ‘{〈0, (1st ‘𝑧)〉, 〈1, (2nd ‘𝑧)〉}) = (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉})) |
16 | 15 | mpompt 7516 | . . . . 5 ⊢ (𝑧 ∈ (𝑋 × 𝑋) ↦ (ℎ‘{〈0, (1st ‘𝑧)〉, 〈1, (2nd ‘𝑧)〉})) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉})) |
17 | 16 | eqcomi 2742 | . . . 4 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉})) = (𝑧 ∈ (𝑋 × 𝑋) ↦ (ℎ‘{〈0, (1st ‘𝑧)〉, 〈1, (2nd ‘𝑧)〉})) |
18 | 7, 17 | fmptd 7108 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ℎ ∈ (2-aryF 𝑋)) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{〈0, 𝑥〉, 〈1, 𝑦〉})):(𝑋 × 𝑋)⟶𝑋) |
19 | sqxpexg 7736 | . . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝑋 × 𝑋) ∈ V) | |
20 | elmapg 8828 | . . . . 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 686 | . . . 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 482 | . . 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 7108 | 1 ⊢ (𝑋 ∈ 𝑉 → 𝐻:(2-aryF 𝑋)⟶(𝑋 ↑m (𝑋 × 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 {cpr 4628 〈cop 4632 ↦ cmpt 5229 × cxp 5672 ⟶wf 6535 ‘cfv 6539 (class class class)co 7403 ∈ cmpo 7405 1st c1st 7967 2nd c2nd 7968 ↑m cmap 8815 0cc0 11105 1c1 11106 2c2 12262 -aryF cnaryf 47213 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-er 8698 df-map 8817 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-nn 12208 df-2 12270 df-n0 12468 df-z 12554 df-uz 12818 df-fz 13480 df-fzo 13623 df-naryf 47214 |
This theorem is referenced by: 2arymaptf1 47240 2arymaptfo 47241 |
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