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 5939 | . 2 ⊢ (〈“𝐼𝐽”〉 “ dom 〈“𝐼𝐽”〉) = ran 〈“𝐼𝐽”〉 | |
2 | s2rn.i | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ 𝐷) | |
3 | s2rn.j | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ 𝐷) | |
4 | 2, 3 | s2cld 14436 | . . . . . 6 ⊢ (𝜑 → 〈“𝐼𝐽”〉 ∈ Word 𝐷) |
5 | wrdfn 14083 | . . . . . 6 ⊢ (〈“𝐼𝐽”〉 ∈ Word 𝐷 → 〈“𝐼𝐽”〉 Fn (0..^(♯‘〈“𝐼𝐽”〉))) | |
6 | s2len 14454 | . . . . . . . . . 10 ⊢ (♯‘〈“𝐼𝐽”〉) = 2 | |
7 | 6 | oveq2i 7224 | . . . . . . . . 9 ⊢ (0..^(♯‘〈“𝐼𝐽”〉)) = (0..^2) |
8 | fzo0to2pr 13327 | . . . . . . . . 9 ⊢ (0..^2) = {0, 1} | |
9 | 7, 8 | eqtri 2765 | . . . . . . . 8 ⊢ (0..^(♯‘〈“𝐼𝐽”〉)) = {0, 1} |
10 | 9 | fneq2i 6477 | . . . . . . 7 ⊢ (〈“𝐼𝐽”〉 Fn (0..^(♯‘〈“𝐼𝐽”〉)) ↔ 〈“𝐼𝐽”〉 Fn {0, 1}) |
11 | 10 | biimpi 219 | . . . . . 6 ⊢ (〈“𝐼𝐽”〉 Fn (0..^(♯‘〈“𝐼𝐽”〉)) → 〈“𝐼𝐽”〉 Fn {0, 1}) |
12 | 4, 5, 11 | 3syl 18 | . . . . 5 ⊢ (𝜑 → 〈“𝐼𝐽”〉 Fn {0, 1}) |
13 | 12 | fndmd 6483 | . . . 4 ⊢ (𝜑 → dom 〈“𝐼𝐽”〉 = {0, 1}) |
14 | 13 | imaeq2d 5929 | . . 3 ⊢ (𝜑 → (〈“𝐼𝐽”〉 “ dom 〈“𝐼𝐽”〉) = (〈“𝐼𝐽”〉 “ {0, 1})) |
15 | c0ex 10827 | . . . . . 6 ⊢ 0 ∈ V | |
16 | 15 | prid1 4678 | . . . . 5 ⊢ 0 ∈ {0, 1} |
17 | 16 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ {0, 1}) |
18 | 1ex 10829 | . . . . . 6 ⊢ 1 ∈ V | |
19 | 18 | prid2 4679 | . . . . 5 ⊢ 1 ∈ {0, 1} |
20 | 19 | a1i 11 | . . . 4 ⊢ (𝜑 → 1 ∈ {0, 1}) |
21 | fnimapr 6795 | . . . 4 ⊢ ((〈“𝐼𝐽”〉 Fn {0, 1} ∧ 0 ∈ {0, 1} ∧ 1 ∈ {0, 1}) → (〈“𝐼𝐽”〉 “ {0, 1}) = {(〈“𝐼𝐽”〉‘0), (〈“𝐼𝐽”〉‘1)}) | |
22 | 12, 17, 20, 21 | syl3anc 1373 | . . 3 ⊢ (𝜑 → (〈“𝐼𝐽”〉 “ {0, 1}) = {(〈“𝐼𝐽”〉‘0), (〈“𝐼𝐽”〉‘1)}) |
23 | s2fv0 14452 | . . . . 5 ⊢ (𝐼 ∈ 𝐷 → (〈“𝐼𝐽”〉‘0) = 𝐼) | |
24 | 2, 23 | syl 17 | . . . 4 ⊢ (𝜑 → (〈“𝐼𝐽”〉‘0) = 𝐼) |
25 | s2fv1 14453 | . . . . 5 ⊢ (𝐽 ∈ 𝐷 → (〈“𝐼𝐽”〉‘1) = 𝐽) | |
26 | 3, 25 | syl 17 | . . . 4 ⊢ (𝜑 → (〈“𝐼𝐽”〉‘1) = 𝐽) |
27 | 24, 26 | preq12d 4657 | . . 3 ⊢ (𝜑 → {(〈“𝐼𝐽”〉‘0), (〈“𝐼𝐽”〉‘1)} = {𝐼, 𝐽}) |
28 | 14, 22, 27 | 3eqtrd 2781 | . 2 ⊢ (𝜑 → (〈“𝐼𝐽”〉 “ dom 〈“𝐼𝐽”〉) = {𝐼, 𝐽}) |
29 | 1, 28 | eqtr3id 2792 | 1 ⊢ (𝜑 → ran 〈“𝐼𝐽”〉 = {𝐼, 𝐽}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 {cpr 4543 dom cdm 5551 ran crn 5552 “ cima 5554 Fn wfn 6375 ‘cfv 6380 (class class class)co 7213 0cc0 10729 1c1 10730 2c2 11885 ..^cfzo 13238 ♯chash 13896 Word cword 14069 〈“cs2 14406 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-n0 12091 df-z 12177 df-uz 12439 df-fz 13096 df-fzo 13239 df-hash 13897 df-word 14070 df-concat 14126 df-s1 14153 df-s2 14413 |
This theorem is referenced by: cycpm2tr 31105 cycpmco2 31119 cyc2fvx 31120 cyc3co2 31126 cyc3conja 31143 |
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