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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 12486 | . . . . . . 7 ⊢ 0 ∈ ℕ0 | |
2 | 1 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℕ0) |
3 | s2f1.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝐷) | |
4 | 1nn0 12487 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
5 | 4 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℕ0) |
6 | s2f1.j | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ 𝐷) | |
7 | 0ne1 12282 | . . . . . . 7 ⊢ 0 ≠ 1 | |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 0 ≠ 1) |
9 | s2f1.1 | . . . . . 6 ⊢ (𝜑 → 𝐼 ≠ 𝐽) | |
10 | f1oprg 6878 | . . . . . . 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 1117 | . . . . . 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 1386 | . . . . 5 ⊢ (𝜑 → {⟨0, 𝐼⟩, ⟨1, 𝐽⟩}:{0, 1}–1-1-onto→{𝐼, 𝐽}) |
13 | s2prop 14857 | . . . . . . 7 ⊢ ((𝐼 ∈ 𝐷 ∧ 𝐽 ∈ 𝐷) → ⟨“𝐼𝐽”⟩ = {⟨0, 𝐼⟩, ⟨1, 𝐽⟩}) | |
14 | 3, 6, 13 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → ⟨“𝐼𝐽”⟩ = {⟨0, 𝐼⟩, ⟨1, 𝐽⟩}) |
15 | 14 | f1oeq1d 6828 | . . . . 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 256 | . . . 4 ⊢ (𝜑 → ⟨“𝐼𝐽”⟩:{0, 1}–1-1-onto→{𝐼, 𝐽}) |
17 | f1of1 6832 | . . . 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 4825 | . . 3 ⊢ (𝜑 → {𝐼, 𝐽} ⊆ 𝐷) |
20 | f1ss 6793 | . . 3 ⊢ ((⟨“𝐼𝐽”⟩:{0, 1}–1-1→{𝐼, 𝐽} ∧ {𝐼, 𝐽} ⊆ 𝐷) → ⟨“𝐼𝐽”⟩:{0, 1}–1-1→𝐷) | |
21 | 18, 19, 20 | syl2anc 584 | . 2 ⊢ (𝜑 → ⟨“𝐼𝐽”⟩:{0, 1}–1-1→𝐷) |
22 | f1dm 6791 | . . . 4 ⊢ (⟨“𝐼𝐽”⟩:{0, 1}–1-1→𝐷 → dom ⟨“𝐼𝐽”⟩ = {0, 1}) | |
23 | 21, 22 | syl 17 | . . 3 ⊢ (𝜑 → dom ⟨“𝐼𝐽”⟩ = {0, 1}) |
24 | f1eq2 6783 | . . 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 256 | 1 ⊢ (𝜑 → ⟨“𝐼𝐽”⟩:dom ⟨“𝐼𝐽”⟩–1-1→𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ⊆ wss 3948 {cpr 4630 ⟨cop 4634 dom cdm 5676 –1-1→wf1 6540 –1-1-onto→wf1o 6542 0cc0 11109 1c1 11110 ℕ0cn0 12471 ⟨“cs2 14791 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-n0 12472 df-z 12558 df-uz 12822 df-fz 13484 df-fzo 13627 df-hash 14290 df-word 14464 df-concat 14520 df-s1 14545 df-s2 14798 |
This theorem is referenced by: cycpm2tr 32273 cycpm2cl 32274 cyc2fv1 32275 cyc2fv2 32276 cycpmco2 32287 cyc2fvx 32288 cyc3co2 32294 cyc3conja 32311 |
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