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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 11900 | . . . . . . 7 ⊢ 0 ∈ ℕ0 | |
2 | 1 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℕ0) |
3 | s2f1.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝐷) | |
4 | 1nn0 11901 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
5 | 4 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℕ0) |
6 | s2f1.j | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ 𝐷) | |
7 | 0ne1 11696 | . . . . . . 7 ⊢ 0 ≠ 1 | |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 0 ≠ 1) |
9 | s2f1.1 | . . . . . 6 ⊢ (𝜑 → 𝐼 ≠ 𝐽) | |
10 | f1oprg 6634 | . . . . . . 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 1114 | . . . . . 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 1383 | . . . . 5 ⊢ (𝜑 → {〈0, 𝐼〉, 〈1, 𝐽〉}:{0, 1}–1-1-onto→{𝐼, 𝐽}) |
13 | s2prop 14260 | . . . . . . 7 ⊢ ((𝐼 ∈ 𝐷 ∧ 𝐽 ∈ 𝐷) → 〈“𝐼𝐽”〉 = {〈0, 𝐼〉, 〈1, 𝐽〉}) | |
14 | 3, 6, 13 | syl2anc 587 | . . . . . 6 ⊢ (𝜑 → 〈“𝐼𝐽”〉 = {〈0, 𝐼〉, 〈1, 𝐽〉}) |
15 | f1oeq1 6579 | . . . . . 6 ⊢ (〈“𝐼𝐽”〉 = {〈0, 𝐼〉, 〈1, 𝐽〉} → (〈“𝐼𝐽”〉:{0, 1}–1-1-onto→{𝐼, 𝐽} ↔ {〈0, 𝐼〉, 〈1, 𝐽〉}:{0, 1}–1-1-onto→{𝐼, 𝐽})) | |
16 | 14, 15 | syl 17 | . . . . 5 ⊢ (𝜑 → (〈“𝐼𝐽”〉:{0, 1}–1-1-onto→{𝐼, 𝐽} ↔ {〈0, 𝐼〉, 〈1, 𝐽〉}:{0, 1}–1-1-onto→{𝐼, 𝐽})) |
17 | 12, 16 | mpbird 260 | . . . 4 ⊢ (𝜑 → 〈“𝐼𝐽”〉:{0, 1}–1-1-onto→{𝐼, 𝐽}) |
18 | f1of1 6589 | . . . 4 ⊢ (〈“𝐼𝐽”〉:{0, 1}–1-1-onto→{𝐼, 𝐽} → 〈“𝐼𝐽”〉:{0, 1}–1-1→{𝐼, 𝐽}) | |
19 | 17, 18 | syl 17 | . . 3 ⊢ (𝜑 → 〈“𝐼𝐽”〉:{0, 1}–1-1→{𝐼, 𝐽}) |
20 | 3, 6 | prssd 4715 | . . 3 ⊢ (𝜑 → {𝐼, 𝐽} ⊆ 𝐷) |
21 | f1ss 6555 | . . 3 ⊢ ((〈“𝐼𝐽”〉:{0, 1}–1-1→{𝐼, 𝐽} ∧ {𝐼, 𝐽} ⊆ 𝐷) → 〈“𝐼𝐽”〉:{0, 1}–1-1→𝐷) | |
22 | 19, 20, 21 | syl2anc 587 | . 2 ⊢ (𝜑 → 〈“𝐼𝐽”〉:{0, 1}–1-1→𝐷) |
23 | f1dm 6553 | . . . 4 ⊢ (〈“𝐼𝐽”〉:{0, 1}–1-1→𝐷 → dom 〈“𝐼𝐽”〉 = {0, 1}) | |
24 | 22, 23 | syl 17 | . . 3 ⊢ (𝜑 → dom 〈“𝐼𝐽”〉 = {0, 1}) |
25 | f1eq2 6545 | . . 3 ⊢ (dom 〈“𝐼𝐽”〉 = {0, 1} → (〈“𝐼𝐽”〉:dom 〈“𝐼𝐽”〉–1-1→𝐷 ↔ 〈“𝐼𝐽”〉:{0, 1}–1-1→𝐷)) | |
26 | 24, 25 | syl 17 | . 2 ⊢ (𝜑 → (〈“𝐼𝐽”〉:dom 〈“𝐼𝐽”〉–1-1→𝐷 ↔ 〈“𝐼𝐽”〉:{0, 1}–1-1→𝐷)) |
27 | 22, 26 | mpbird 260 | 1 ⊢ (𝜑 → 〈“𝐼𝐽”〉:dom 〈“𝐼𝐽”〉–1-1→𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ⊆ wss 3881 {cpr 4527 〈cop 4531 dom cdm 5519 –1-1→wf1 6321 –1-1-onto→wf1o 6323 0cc0 10526 1c1 10527 ℕ0cn0 11885 〈“cs2 14194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-hash 13687 df-word 13858 df-concat 13914 df-s1 13941 df-s2 14201 |
This theorem is referenced by: cycpm2tr 30811 cycpm2cl 30812 cyc2fv1 30813 cyc2fv2 30814 cycpmco2 30825 cyc2fvx 30826 cyc3co2 30832 cyc3conja 30849 |
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