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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2aryenef | Structured version Visualization version GIF version |
Description: The set of binary (endo)functions and the set of binary operations are equinumerous. (Contributed by AV, 19-May-2024.) |
Ref | Expression |
---|---|
2aryenef | ⊢ (2-aryF 𝑋) ≈ (𝑋 ↑m (𝑋 × 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 7442 | . . . . . 6 ⊢ (2-aryF 𝑋) ∈ V | |
2 | 1 | mptex 7225 | . . . . 5 ⊢ (𝑓 ∈ (2-aryF 𝑋) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑓‘{⟨0, 𝑥⟩, ⟨1, 𝑦⟩}))) ∈ V |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝑋 ∈ V → (𝑓 ∈ (2-aryF 𝑋) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑓‘{⟨0, 𝑥⟩, ⟨1, 𝑦⟩}))) ∈ V) |
4 | eqid 2733 | . . . . 5 ⊢ (𝑓 ∈ (2-aryF 𝑋) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑓‘{⟨0, 𝑥⟩, ⟨1, 𝑦⟩}))) = (𝑓 ∈ (2-aryF 𝑋) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑓‘{⟨0, 𝑥⟩, ⟨1, 𝑦⟩}))) | |
5 | 4 | 2arymaptf1o 47341 | . . . 4 ⊢ (𝑋 ∈ V → (𝑓 ∈ (2-aryF 𝑋) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑓‘{⟨0, 𝑥⟩, ⟨1, 𝑦⟩}))):(2-aryF 𝑋)–1-1-onto→(𝑋 ↑m (𝑋 × 𝑋))) |
6 | f1oeq1 6822 | . . . 4 ⊢ (ℎ = (𝑓 ∈ (2-aryF 𝑋) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑓‘{⟨0, 𝑥⟩, ⟨1, 𝑦⟩}))) → (ℎ:(2-aryF 𝑋)–1-1-onto→(𝑋 ↑m (𝑋 × 𝑋)) ↔ (𝑓 ∈ (2-aryF 𝑋) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑓‘{⟨0, 𝑥⟩, ⟨1, 𝑦⟩}))):(2-aryF 𝑋)–1-1-onto→(𝑋 ↑m (𝑋 × 𝑋)))) | |
7 | 3, 5, 6 | spcedv 3589 | . . 3 ⊢ (𝑋 ∈ V → ∃ℎ ℎ:(2-aryF 𝑋)–1-1-onto→(𝑋 ↑m (𝑋 × 𝑋))) |
8 | bren 8949 | . . 3 ⊢ ((2-aryF 𝑋) ≈ (𝑋 ↑m (𝑋 × 𝑋)) ↔ ∃ℎ ℎ:(2-aryF 𝑋)–1-1-onto→(𝑋 ↑m (𝑋 × 𝑋))) | |
9 | 7, 8 | sylibr 233 | . 2 ⊢ (𝑋 ∈ V → (2-aryF 𝑋) ≈ (𝑋 ↑m (𝑋 × 𝑋))) |
10 | 0ex 5308 | . . . . 5 ⊢ ∅ ∈ V | |
11 | 10 | enref 8981 | . . . 4 ⊢ ∅ ≈ ∅ |
12 | 11 | a1i 11 | . . 3 ⊢ (¬ 𝑋 ∈ V → ∅ ≈ ∅) |
13 | df-naryf 47313 | . . . . 5 ⊢ -aryF = (𝑛 ∈ ℕ0, 𝑥 ∈ V ↦ (𝑥 ↑m (𝑥 ↑m (0..^𝑛)))) | |
14 | 13 | reldmmpo 7543 | . . . 4 ⊢ Rel dom -aryF |
15 | 14 | ovprc2 7449 | . . 3 ⊢ (¬ 𝑋 ∈ V → (2-aryF 𝑋) = ∅) |
16 | reldmmap 8829 | . . . 4 ⊢ Rel dom ↑m | |
17 | 16 | ovprc1 7448 | . . 3 ⊢ (¬ 𝑋 ∈ V → (𝑋 ↑m (𝑋 × 𝑋)) = ∅) |
18 | 12, 15, 17 | 3brtr4d 5181 | . 2 ⊢ (¬ 𝑋 ∈ V → (2-aryF 𝑋) ≈ (𝑋 ↑m (𝑋 × 𝑋))) |
19 | 9, 18 | pm2.61i 182 | 1 ⊢ (2-aryF 𝑋) ≈ (𝑋 ↑m (𝑋 × 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∃wex 1782 ∈ wcel 2107 Vcvv 3475 ∅c0 4323 {cpr 4631 ⟨cop 4635 class class class wbr 5149 ↦ cmpt 5232 × cxp 5675 –1-1-onto→wf1o 6543 ‘cfv 6544 (class class class)co 7409 ∈ cmpo 7411 ↑m cmap 8820 ≈ cen 8936 0cc0 11110 1c1 11111 2c2 12267 ℕ0cn0 12472 ..^cfzo 13627 -aryF cnaryf 47312 |
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-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
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 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485 df-fzo 13628 df-naryf 47313 |
This theorem is referenced by: (None) |
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