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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > s2rnOLD | Structured version Visualization version GIF version |
Description: Obsolete version of s2rn 15008 as of 1-Aug-2025. Range of a length 2 string. (Contributed by Thierry Arnoux, 19-Sep-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
s2rnOLD.i | ⊢ (𝜑 → 𝐼 ∈ 𝐷) |
s2rnOLD.j | ⊢ (𝜑 → 𝐽 ∈ 𝐷) |
Ref | Expression |
---|---|
s2rnOLD | ⊢ (𝜑 → ran 〈“𝐼𝐽”〉 = {𝐼, 𝐽}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imadmrn 6098 | . 2 ⊢ (〈“𝐼𝐽”〉 “ dom 〈“𝐼𝐽”〉) = ran 〈“𝐼𝐽”〉 | |
2 | s2rnOLD.i | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ 𝐷) | |
3 | s2rnOLD.j | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ 𝐷) | |
4 | 2, 3 | s2cld 14916 | . . . . . 6 ⊢ (𝜑 → 〈“𝐼𝐽”〉 ∈ Word 𝐷) |
5 | wrdfn 14572 | . . . . . 6 ⊢ (〈“𝐼𝐽”〉 ∈ Word 𝐷 → 〈“𝐼𝐽”〉 Fn (0..^(♯‘〈“𝐼𝐽”〉))) | |
6 | s2len 14934 | . . . . . . . . . 10 ⊢ (♯‘〈“𝐼𝐽”〉) = 2 | |
7 | 6 | oveq2i 7456 | . . . . . . . . 9 ⊢ (0..^(♯‘〈“𝐼𝐽”〉)) = (0..^2) |
8 | fzo0to2pr 13797 | . . . . . . . . 9 ⊢ (0..^2) = {0, 1} | |
9 | 7, 8 | eqtri 2762 | . . . . . . . 8 ⊢ (0..^(♯‘〈“𝐼𝐽”〉)) = {0, 1} |
10 | 9 | fneq2i 6676 | . . . . . . 7 ⊢ (〈“𝐼𝐽”〉 Fn (0..^(♯‘〈“𝐼𝐽”〉)) ↔ 〈“𝐼𝐽”〉 Fn {0, 1}) |
11 | 10 | biimpi 216 | . . . . . 6 ⊢ (〈“𝐼𝐽”〉 Fn (0..^(♯‘〈“𝐼𝐽”〉)) → 〈“𝐼𝐽”〉 Fn {0, 1}) |
12 | 4, 5, 11 | 3syl 18 | . . . . 5 ⊢ (𝜑 → 〈“𝐼𝐽”〉 Fn {0, 1}) |
13 | 12 | fndmd 6683 | . . . 4 ⊢ (𝜑 → dom 〈“𝐼𝐽”〉 = {0, 1}) |
14 | 13 | imaeq2d 6088 | . . 3 ⊢ (𝜑 → (〈“𝐼𝐽”〉 “ dom 〈“𝐼𝐽”〉) = (〈“𝐼𝐽”〉 “ {0, 1})) |
15 | c0ex 11280 | . . . . . 6 ⊢ 0 ∈ V | |
16 | 15 | prid1 4787 | . . . . 5 ⊢ 0 ∈ {0, 1} |
17 | 16 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ {0, 1}) |
18 | 1ex 11282 | . . . . . 6 ⊢ 1 ∈ V | |
19 | 18 | prid2 4788 | . . . . 5 ⊢ 1 ∈ {0, 1} |
20 | 19 | a1i 11 | . . . 4 ⊢ (𝜑 → 1 ∈ {0, 1}) |
21 | fnimapr 7003 | . . . 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 14932 | . . . . 5 ⊢ (𝐼 ∈ 𝐷 → (〈“𝐼𝐽”〉‘0) = 𝐼) | |
24 | 2, 23 | syl 17 | . . . 4 ⊢ (𝜑 → (〈“𝐼𝐽”〉‘0) = 𝐼) |
25 | s2fv1 14933 | . . . . 5 ⊢ (𝐽 ∈ 𝐷 → (〈“𝐼𝐽”〉‘1) = 𝐽) | |
26 | 3, 25 | syl 17 | . . . 4 ⊢ (𝜑 → (〈“𝐼𝐽”〉‘1) = 𝐽) |
27 | 24, 26 | preq12d 4766 | . . 3 ⊢ (𝜑 → {(〈“𝐼𝐽”〉‘0), (〈“𝐼𝐽”〉‘1)} = {𝐼, 𝐽}) |
28 | 14, 22, 27 | 3eqtrd 2778 | . 2 ⊢ (𝜑 → (〈“𝐼𝐽”〉 “ dom 〈“𝐼𝐽”〉) = {𝐼, 𝐽}) |
29 | 1, 28 | eqtr3id 2788 | 1 ⊢ (𝜑 → ran 〈“𝐼𝐽”〉 = {𝐼, 𝐽}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2103 {cpr 4650 dom cdm 5699 ran crn 5700 “ cima 5702 Fn wfn 6567 ‘cfv 6572 (class class class)co 7445 0cc0 11180 1c1 11181 2c2 12344 ..^cfzo 13707 ♯chash 14375 Word cword 14558 〈“cs2 14886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-1st 8026 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-er 8759 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-card 10004 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-nn 12290 df-2 12352 df-n0 12550 df-z 12636 df-uz 12900 df-fz 13564 df-fzo 13708 df-hash 14376 df-word 14559 df-concat 14615 df-s1 14640 df-s2 14893 |
This theorem is referenced by: (None) |
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