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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > s2rn | Structured version Visualization version GIF version |
Description: Range of a length 2 string. (Contributed by Thierry Arnoux, 19-Sep-2023.) |
Ref | Expression |
---|---|
s2rn.i | ⊢ (𝜑 → 𝐼 ∈ 𝐷) |
s2rn.j | ⊢ (𝜑 → 𝐽 ∈ 𝐷) |
Ref | Expression |
---|---|
s2rn | ⊢ (𝜑 → ran ⟨“𝐼𝐽”⟩ = {𝐼, 𝐽}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imadmrn 6067 | . 2 ⊢ (⟨“𝐼𝐽”⟩ “ dom ⟨“𝐼𝐽”⟩) = ran ⟨“𝐼𝐽”⟩ | |
2 | s2rn.i | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ 𝐷) | |
3 | s2rn.j | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ 𝐷) | |
4 | 2, 3 | s2cld 14818 | . . . . . 6 ⊢ (𝜑 → ⟨“𝐼𝐽”⟩ ∈ Word 𝐷) |
5 | wrdfn 14474 | . . . . . 6 ⊢ (⟨“𝐼𝐽”⟩ ∈ Word 𝐷 → ⟨“𝐼𝐽”⟩ Fn (0..^(♯‘⟨“𝐼𝐽”⟩))) | |
6 | s2len 14836 | . . . . . . . . . 10 ⊢ (♯‘⟨“𝐼𝐽”⟩) = 2 | |
7 | 6 | oveq2i 7416 | . . . . . . . . 9 ⊢ (0..^(♯‘⟨“𝐼𝐽”⟩)) = (0..^2) |
8 | fzo0to2pr 13713 | . . . . . . . . 9 ⊢ (0..^2) = {0, 1} | |
9 | 7, 8 | eqtri 2760 | . . . . . . . 8 ⊢ (0..^(♯‘⟨“𝐼𝐽”⟩)) = {0, 1} |
10 | 9 | fneq2i 6644 | . . . . . . 7 ⊢ (⟨“𝐼𝐽”⟩ Fn (0..^(♯‘⟨“𝐼𝐽”⟩)) ↔ ⟨“𝐼𝐽”⟩ Fn {0, 1}) |
11 | 10 | biimpi 215 | . . . . . 6 ⊢ (⟨“𝐼𝐽”⟩ Fn (0..^(♯‘⟨“𝐼𝐽”⟩)) → ⟨“𝐼𝐽”⟩ Fn {0, 1}) |
12 | 4, 5, 11 | 3syl 18 | . . . . 5 ⊢ (𝜑 → ⟨“𝐼𝐽”⟩ Fn {0, 1}) |
13 | 12 | fndmd 6651 | . . . 4 ⊢ (𝜑 → dom ⟨“𝐼𝐽”⟩ = {0, 1}) |
14 | 13 | imaeq2d 6057 | . . 3 ⊢ (𝜑 → (⟨“𝐼𝐽”⟩ “ dom ⟨“𝐼𝐽”⟩) = (⟨“𝐼𝐽”⟩ “ {0, 1})) |
15 | c0ex 11204 | . . . . . 6 ⊢ 0 ∈ V | |
16 | 15 | prid1 4765 | . . . . 5 ⊢ 0 ∈ {0, 1} |
17 | 16 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ {0, 1}) |
18 | 1ex 11206 | . . . . . 6 ⊢ 1 ∈ V | |
19 | 18 | prid2 4766 | . . . . 5 ⊢ 1 ∈ {0, 1} |
20 | 19 | a1i 11 | . . . 4 ⊢ (𝜑 → 1 ∈ {0, 1}) |
21 | fnimapr 6972 | . . . 4 ⊢ ((⟨“𝐼𝐽”⟩ Fn {0, 1} ∧ 0 ∈ {0, 1} ∧ 1 ∈ {0, 1}) → (⟨“𝐼𝐽”⟩ “ {0, 1}) = {(⟨“𝐼𝐽”⟩‘0), (⟨“𝐼𝐽”⟩‘1)}) | |
22 | 12, 17, 20, 21 | syl3anc 1371 | . . 3 ⊢ (𝜑 → (⟨“𝐼𝐽”⟩ “ {0, 1}) = {(⟨“𝐼𝐽”⟩‘0), (⟨“𝐼𝐽”⟩‘1)}) |
23 | s2fv0 14834 | . . . . 5 ⊢ (𝐼 ∈ 𝐷 → (⟨“𝐼𝐽”⟩‘0) = 𝐼) | |
24 | 2, 23 | syl 17 | . . . 4 ⊢ (𝜑 → (⟨“𝐼𝐽”⟩‘0) = 𝐼) |
25 | s2fv1 14835 | . . . . 5 ⊢ (𝐽 ∈ 𝐷 → (⟨“𝐼𝐽”⟩‘1) = 𝐽) | |
26 | 3, 25 | syl 17 | . . . 4 ⊢ (𝜑 → (⟨“𝐼𝐽”⟩‘1) = 𝐽) |
27 | 24, 26 | preq12d 4744 | . . 3 ⊢ (𝜑 → {(⟨“𝐼𝐽”⟩‘0), (⟨“𝐼𝐽”⟩‘1)} = {𝐼, 𝐽}) |
28 | 14, 22, 27 | 3eqtrd 2776 | . 2 ⊢ (𝜑 → (⟨“𝐼𝐽”⟩ “ dom ⟨“𝐼𝐽”⟩) = {𝐼, 𝐽}) |
29 | 1, 28 | eqtr3id 2786 | 1 ⊢ (𝜑 → ran ⟨“𝐼𝐽”⟩ = {𝐼, 𝐽}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 {cpr 4629 dom cdm 5675 ran crn 5676 “ cima 5678 Fn wfn 6535 ‘cfv 6540 (class class class)co 7405 0cc0 11106 1c1 11107 2c2 12263 ..^cfzo 13623 ♯chash 14286 Word cword 14460 ⟨“cs2 14788 |
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 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
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 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-fzo 13624 df-hash 14287 df-word 14461 df-concat 14517 df-s1 14542 df-s2 14795 |
This theorem is referenced by: cycpm2tr 32265 cycpmco2 32279 cyc2fvx 32280 cyc3co2 32286 cyc3conja 32303 |
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