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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 6024 | . 2 ⊢ (⟨“𝐼𝐽”⟩ “ dom ⟨“𝐼𝐽”⟩) = ran ⟨“𝐼𝐽”⟩ | |
2 | s2rn.i | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ 𝐷) | |
3 | s2rn.j | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ 𝐷) | |
4 | 2, 3 | s2cld 14766 | . . . . . 6 ⊢ (𝜑 → ⟨“𝐼𝐽”⟩ ∈ Word 𝐷) |
5 | wrdfn 14422 | . . . . . 6 ⊢ (⟨“𝐼𝐽”⟩ ∈ Word 𝐷 → ⟨“𝐼𝐽”⟩ Fn (0..^(♯‘⟨“𝐼𝐽”⟩))) | |
6 | s2len 14784 | . . . . . . . . . 10 ⊢ (♯‘⟨“𝐼𝐽”⟩) = 2 | |
7 | 6 | oveq2i 7369 | . . . . . . . . 9 ⊢ (0..^(♯‘⟨“𝐼𝐽”⟩)) = (0..^2) |
8 | fzo0to2pr 13663 | . . . . . . . . 9 ⊢ (0..^2) = {0, 1} | |
9 | 7, 8 | eqtri 2761 | . . . . . . . 8 ⊢ (0..^(♯‘⟨“𝐼𝐽”⟩)) = {0, 1} |
10 | 9 | fneq2i 6601 | . . . . . . 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 6608 | . . . 4 ⊢ (𝜑 → dom ⟨“𝐼𝐽”⟩ = {0, 1}) |
14 | 13 | imaeq2d 6014 | . . 3 ⊢ (𝜑 → (⟨“𝐼𝐽”⟩ “ dom ⟨“𝐼𝐽”⟩) = (⟨“𝐼𝐽”⟩ “ {0, 1})) |
15 | c0ex 11154 | . . . . . 6 ⊢ 0 ∈ V | |
16 | 15 | prid1 4724 | . . . . 5 ⊢ 0 ∈ {0, 1} |
17 | 16 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ {0, 1}) |
18 | 1ex 11156 | . . . . . 6 ⊢ 1 ∈ V | |
19 | 18 | prid2 4725 | . . . . 5 ⊢ 1 ∈ {0, 1} |
20 | 19 | a1i 11 | . . . 4 ⊢ (𝜑 → 1 ∈ {0, 1}) |
21 | fnimapr 6926 | . . . 4 ⊢ ((⟨“𝐼𝐽”⟩ Fn {0, 1} ∧ 0 ∈ {0, 1} ∧ 1 ∈ {0, 1}) → (⟨“𝐼𝐽”⟩ “ {0, 1}) = {(⟨“𝐼𝐽”⟩‘0), (⟨“𝐼𝐽”⟩‘1)}) | |
22 | 12, 17, 20, 21 | syl3anc 1372 | . . 3 ⊢ (𝜑 → (⟨“𝐼𝐽”⟩ “ {0, 1}) = {(⟨“𝐼𝐽”⟩‘0), (⟨“𝐼𝐽”⟩‘1)}) |
23 | s2fv0 14782 | . . . . 5 ⊢ (𝐼 ∈ 𝐷 → (⟨“𝐼𝐽”⟩‘0) = 𝐼) | |
24 | 2, 23 | syl 17 | . . . 4 ⊢ (𝜑 → (⟨“𝐼𝐽”⟩‘0) = 𝐼) |
25 | s2fv1 14783 | . . . . 5 ⊢ (𝐽 ∈ 𝐷 → (⟨“𝐼𝐽”⟩‘1) = 𝐽) | |
26 | 3, 25 | syl 17 | . . . 4 ⊢ (𝜑 → (⟨“𝐼𝐽”⟩‘1) = 𝐽) |
27 | 24, 26 | preq12d 4703 | . . 3 ⊢ (𝜑 → {(⟨“𝐼𝐽”⟩‘0), (⟨“𝐼𝐽”⟩‘1)} = {𝐼, 𝐽}) |
28 | 14, 22, 27 | 3eqtrd 2777 | . 2 ⊢ (𝜑 → (⟨“𝐼𝐽”⟩ “ dom ⟨“𝐼𝐽”⟩) = {𝐼, 𝐽}) |
29 | 1, 28 | eqtr3id 2787 | 1 ⊢ (𝜑 → ran ⟨“𝐼𝐽”⟩ = {𝐼, 𝐽}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 {cpr 4589 dom cdm 5634 ran crn 5635 “ cima 5637 Fn wfn 6492 ‘cfv 6497 (class class class)co 7358 0cc0 11056 1c1 11057 2c2 12213 ..^cfzo 13573 ♯chash 14236 Word cword 14408 ⟨“cs2 14736 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-n0 12419 df-z 12505 df-uz 12769 df-fz 13431 df-fzo 13574 df-hash 14237 df-word 14409 df-concat 14465 df-s1 14490 df-s2 14743 |
This theorem is referenced by: cycpm2tr 32017 cycpmco2 32031 cyc2fvx 32032 cyc3co2 32038 cyc3conja 32055 |
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