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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > s2f1 | Structured version Visualization version GIF version |
Description: Conditions for a length 2 string to be a one-to-one function. (Contributed by Thierry Arnoux, 19-Sep-2023.) |
Ref | Expression |
---|---|
s2f1.i | ⊢ (𝜑 → 𝐼 ∈ 𝐷) |
s2f1.j | ⊢ (𝜑 → 𝐽 ∈ 𝐷) |
s2f1.1 | ⊢ (𝜑 → 𝐼 ≠ 𝐽) |
Ref | Expression |
---|---|
s2f1 | ⊢ (𝜑 → ⟨“𝐼𝐽”⟩:dom ⟨“𝐼𝐽”⟩–1-1→𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nn0 12436 | . . . . . . 7 ⊢ 0 ∈ ℕ0 | |
2 | 1 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℕ0) |
3 | s2f1.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝐷) | |
4 | 1nn0 12437 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
5 | 4 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℕ0) |
6 | s2f1.j | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ 𝐷) | |
7 | 0ne1 12232 | . . . . . . 7 ⊢ 0 ≠ 1 | |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 0 ≠ 1) |
9 | s2f1.1 | . . . . . 6 ⊢ (𝜑 → 𝐼 ≠ 𝐽) | |
10 | f1oprg 6833 | . . . . . . 7 ⊢ (((0 ∈ ℕ0 ∧ 𝐼 ∈ 𝐷) ∧ (1 ∈ ℕ0 ∧ 𝐽 ∈ 𝐷)) → ((0 ≠ 1 ∧ 𝐼 ≠ 𝐽) → {⟨0, 𝐼⟩, ⟨1, 𝐽⟩}:{0, 1}–1-1-onto→{𝐼, 𝐽})) | |
11 | 10 | 3impia 1118 | . . . . . 6 ⊢ (((0 ∈ ℕ0 ∧ 𝐼 ∈ 𝐷) ∧ (1 ∈ ℕ0 ∧ 𝐽 ∈ 𝐷) ∧ (0 ≠ 1 ∧ 𝐼 ≠ 𝐽)) → {⟨0, 𝐼⟩, ⟨1, 𝐽⟩}:{0, 1}–1-1-onto→{𝐼, 𝐽}) |
12 | 2, 3, 5, 6, 8, 9, 11 | syl222anc 1387 | . . . . 5 ⊢ (𝜑 → {⟨0, 𝐼⟩, ⟨1, 𝐽⟩}:{0, 1}–1-1-onto→{𝐼, 𝐽}) |
13 | s2prop 14805 | . . . . . . 7 ⊢ ((𝐼 ∈ 𝐷 ∧ 𝐽 ∈ 𝐷) → ⟨“𝐼𝐽”⟩ = {⟨0, 𝐼⟩, ⟨1, 𝐽⟩}) | |
14 | 3, 6, 13 | syl2anc 585 | . . . . . 6 ⊢ (𝜑 → ⟨“𝐼𝐽”⟩ = {⟨0, 𝐼⟩, ⟨1, 𝐽⟩}) |
15 | 14 | f1oeq1d 6783 | . . . . 5 ⊢ (𝜑 → (⟨“𝐼𝐽”⟩:{0, 1}–1-1-onto→{𝐼, 𝐽} ↔ {⟨0, 𝐼⟩, ⟨1, 𝐽⟩}:{0, 1}–1-1-onto→{𝐼, 𝐽})) |
16 | 12, 15 | mpbird 257 | . . . 4 ⊢ (𝜑 → ⟨“𝐼𝐽”⟩:{0, 1}–1-1-onto→{𝐼, 𝐽}) |
17 | f1of1 6787 | . . . 4 ⊢ (⟨“𝐼𝐽”⟩:{0, 1}–1-1-onto→{𝐼, 𝐽} → ⟨“𝐼𝐽”⟩:{0, 1}–1-1→{𝐼, 𝐽}) | |
18 | 16, 17 | syl 17 | . . 3 ⊢ (𝜑 → ⟨“𝐼𝐽”⟩:{0, 1}–1-1→{𝐼, 𝐽}) |
19 | 3, 6 | prssd 4786 | . . 3 ⊢ (𝜑 → {𝐼, 𝐽} ⊆ 𝐷) |
20 | f1ss 6748 | . . 3 ⊢ ((⟨“𝐼𝐽”⟩:{0, 1}–1-1→{𝐼, 𝐽} ∧ {𝐼, 𝐽} ⊆ 𝐷) → ⟨“𝐼𝐽”⟩:{0, 1}–1-1→𝐷) | |
21 | 18, 19, 20 | syl2anc 585 | . 2 ⊢ (𝜑 → ⟨“𝐼𝐽”⟩:{0, 1}–1-1→𝐷) |
22 | f1dm 6746 | . . . 4 ⊢ (⟨“𝐼𝐽”⟩:{0, 1}–1-1→𝐷 → dom ⟨“𝐼𝐽”⟩ = {0, 1}) | |
23 | 21, 22 | syl 17 | . . 3 ⊢ (𝜑 → dom ⟨“𝐼𝐽”⟩ = {0, 1}) |
24 | f1eq2 6738 | . . 3 ⊢ (dom ⟨“𝐼𝐽”⟩ = {0, 1} → (⟨“𝐼𝐽”⟩:dom ⟨“𝐼𝐽”⟩–1-1→𝐷 ↔ ⟨“𝐼𝐽”⟩:{0, 1}–1-1→𝐷)) | |
25 | 23, 24 | syl 17 | . 2 ⊢ (𝜑 → (⟨“𝐼𝐽”⟩:dom ⟨“𝐼𝐽”⟩–1-1→𝐷 ↔ ⟨“𝐼𝐽”⟩:{0, 1}–1-1→𝐷)) |
26 | 21, 25 | mpbird 257 | 1 ⊢ (𝜑 → ⟨“𝐼𝐽”⟩:dom ⟨“𝐼𝐽”⟩–1-1→𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 ⊆ wss 3914 {cpr 4592 ⟨cop 4596 dom cdm 5637 –1-1→wf1 6497 –1-1-onto→wf1o 6499 0cc0 11059 1c1 11060 ℕ0cn0 12421 ⟨“cs2 14739 |
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 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 df-fzo 13577 df-hash 14240 df-word 14412 df-concat 14468 df-s1 14493 df-s2 14746 |
This theorem is referenced by: cycpm2tr 32024 cycpm2cl 32025 cyc2fv1 32026 cyc2fv2 32027 cycpmco2 32038 cyc2fvx 32039 cyc3co2 32045 cyc3conja 32062 |
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