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 11913 | . . . . . . 7 ⊢ 0 ∈ ℕ0 | |
2 | 1 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℕ0) |
3 | s2f1.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝐷) | |
4 | 1nn0 11914 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
5 | 4 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℕ0) |
6 | s2f1.j | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ 𝐷) | |
7 | 0ne1 11709 | . . . . . . 7 ⊢ 0 ≠ 1 | |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 0 ≠ 1) |
9 | s2f1.1 | . . . . . 6 ⊢ (𝜑 → 𝐼 ≠ 𝐽) | |
10 | f1oprg 6659 | . . . . . . 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 1113 | . . . . . 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 1382 | . . . . 5 ⊢ (𝜑 → {〈0, 𝐼〉, 〈1, 𝐽〉}:{0, 1}–1-1-onto→{𝐼, 𝐽}) |
13 | s2prop 14269 | . . . . . . 7 ⊢ ((𝐼 ∈ 𝐷 ∧ 𝐽 ∈ 𝐷) → 〈“𝐼𝐽”〉 = {〈0, 𝐼〉, 〈1, 𝐽〉}) | |
14 | 3, 6, 13 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → 〈“𝐼𝐽”〉 = {〈0, 𝐼〉, 〈1, 𝐽〉}) |
15 | f1oeq1 6604 | . . . . . 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 259 | . . . 4 ⊢ (𝜑 → 〈“𝐼𝐽”〉:{0, 1}–1-1-onto→{𝐼, 𝐽}) |
18 | f1of1 6614 | . . . 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 4755 | . . 3 ⊢ (𝜑 → {𝐼, 𝐽} ⊆ 𝐷) |
21 | f1ss 6580 | . . 3 ⊢ ((〈“𝐼𝐽”〉:{0, 1}–1-1→{𝐼, 𝐽} ∧ {𝐼, 𝐽} ⊆ 𝐷) → 〈“𝐼𝐽”〉:{0, 1}–1-1→𝐷) | |
22 | 19, 20, 21 | syl2anc 586 | . 2 ⊢ (𝜑 → 〈“𝐼𝐽”〉:{0, 1}–1-1→𝐷) |
23 | f1dm 6579 | . . . 4 ⊢ (〈“𝐼𝐽”〉:{0, 1}–1-1→𝐷 → dom 〈“𝐼𝐽”〉 = {0, 1}) | |
24 | 22, 23 | syl 17 | . . 3 ⊢ (𝜑 → dom 〈“𝐼𝐽”〉 = {0, 1}) |
25 | f1eq2 6571 | . . 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 259 | 1 ⊢ (𝜑 → 〈“𝐼𝐽”〉:dom 〈“𝐼𝐽”〉–1-1→𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ⊆ wss 3936 {cpr 4569 〈cop 4573 dom cdm 5555 –1-1→wf1 6352 –1-1-onto→wf1o 6354 0cc0 10537 1c1 10538 ℕ0cn0 11898 〈“cs2 14203 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-hash 13692 df-word 13863 df-concat 13923 df-s1 13950 df-s2 14210 |
This theorem is referenced by: cycpm2tr 30761 cycpm2cl 30762 cyc2fv1 30763 cyc2fv2 30764 cycpmco2 30775 cyc2fvx 30776 cyc3co2 30782 cyc3conja 30799 |
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