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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > s1f1 | Structured version Visualization version GIF version |
Description: Conditions for a length 1 string to be a one-to-one function. (Contributed by Thierry Arnoux, 11-Dec-2023.) |
Ref | Expression |
---|---|
s1f1.1 | ⊢ (𝜑 → 𝐼 ∈ 𝐷) |
Ref | Expression |
---|---|
s1f1 | ⊢ (𝜑 → ⟨“𝐼”⟩:dom ⟨“𝐼”⟩–1-1→𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nn0 12485 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (𝜑 → 0 ∈ ℕ0) |
3 | s1f1.1 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝐷) | |
4 | f1osng 6865 | . . . . 5 ⊢ ((0 ∈ ℕ0 ∧ 𝐼 ∈ 𝐷) → {⟨0, 𝐼⟩}:{0}–1-1-onto→{𝐼}) | |
5 | 2, 3, 4 | syl2anc 583 | . . . 4 ⊢ (𝜑 → {⟨0, 𝐼⟩}:{0}–1-1-onto→{𝐼}) |
6 | f1of1 6823 | . . . 4 ⊢ ({⟨0, 𝐼⟩}:{0}–1-1-onto→{𝐼} → {⟨0, 𝐼⟩}:{0}–1-1→{𝐼}) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝜑 → {⟨0, 𝐼⟩}:{0}–1-1→{𝐼}) |
8 | 3 | snssd 4805 | . . 3 ⊢ (𝜑 → {𝐼} ⊆ 𝐷) |
9 | f1ss 6784 | . . 3 ⊢ (({⟨0, 𝐼⟩}:{0}–1-1→{𝐼} ∧ {𝐼} ⊆ 𝐷) → {⟨0, 𝐼⟩}:{0}–1-1→𝐷) | |
10 | 7, 8, 9 | syl2anc 583 | . 2 ⊢ (𝜑 → {⟨0, 𝐼⟩}:{0}–1-1→𝐷) |
11 | s1val 14546 | . . . 4 ⊢ (𝐼 ∈ 𝐷 → ⟨“𝐼”⟩ = {⟨0, 𝐼⟩}) | |
12 | 3, 11 | syl 17 | . . 3 ⊢ (𝜑 → ⟨“𝐼”⟩ = {⟨0, 𝐼⟩}) |
13 | s1dm 14556 | . . . 4 ⊢ dom ⟨“𝐼”⟩ = {0} | |
14 | 13 | a1i 11 | . . 3 ⊢ (𝜑 → dom ⟨“𝐼”⟩ = {0}) |
15 | eqidd 2725 | . . 3 ⊢ (𝜑 → 𝐷 = 𝐷) | |
16 | 12, 14, 15 | f1eq123d 6816 | . 2 ⊢ (𝜑 → (⟨“𝐼”⟩:dom ⟨“𝐼”⟩–1-1→𝐷 ↔ {⟨0, 𝐼⟩}:{0}–1-1→𝐷)) |
17 | 10, 16 | mpbird 257 | 1 ⊢ (𝜑 → ⟨“𝐼”⟩:dom ⟨“𝐼”⟩–1-1→𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⊆ wss 3941 {csn 4621 ⟨cop 4627 dom cdm 5667 –1-1→wf1 6531 –1-1-onto→wf1o 6533 0cc0 11107 ℕ0cn0 12470 ⟨“cs1 14543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-card 9931 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-n0 12471 df-z 12557 df-uz 12821 df-fz 13483 df-fzo 13626 df-hash 14289 df-word 14463 df-s1 14544 |
This theorem is referenced by: cycpmco2f1 32754 |
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