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Mirrors > Home > MPE Home > Th. List > s2dm | Structured version Visualization version GIF version |
Description: The domain of a doubleton word is an unordered pair. (Contributed by AV, 9-Jan-2020.) |
Ref | Expression |
---|---|
s2dm | ⊢ dom 〈“𝐴𝐵”〉 = {0, 1} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | s2cli 14098 | . . . 4 ⊢ 〈“𝐴𝐵”〉 ∈ Word V | |
2 | wrdf 13671 | . . . 4 ⊢ (〈“𝐴𝐵”〉 ∈ Word V → 〈“𝐴𝐵”〉:(0..^(♯‘〈“𝐴𝐵”〉))⟶V) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ 〈“𝐴𝐵”〉:(0..^(♯‘〈“𝐴𝐵”〉))⟶V |
4 | s2len 14107 | . . . . 5 ⊢ (♯‘〈“𝐴𝐵”〉) = 2 | |
5 | oveq2 6978 | . . . . . 6 ⊢ ((♯‘〈“𝐴𝐵”〉) = 2 → (0..^(♯‘〈“𝐴𝐵”〉)) = (0..^2)) | |
6 | fzo0to2pr 12931 | . . . . . 6 ⊢ (0..^2) = {0, 1} | |
7 | 5, 6 | syl6eq 2824 | . . . . 5 ⊢ ((♯‘〈“𝐴𝐵”〉) = 2 → (0..^(♯‘〈“𝐴𝐵”〉)) = {0, 1}) |
8 | 4, 7 | ax-mp 5 | . . . 4 ⊢ (0..^(♯‘〈“𝐴𝐵”〉)) = {0, 1} |
9 | 8 | feq2i 6330 | . . 3 ⊢ (〈“𝐴𝐵”〉:(0..^(♯‘〈“𝐴𝐵”〉))⟶V ↔ 〈“𝐴𝐵”〉:{0, 1}⟶V) |
10 | 3, 9 | mpbi 222 | . 2 ⊢ 〈“𝐴𝐵”〉:{0, 1}⟶V |
11 | 10 | fdmi 6348 | 1 ⊢ dom 〈“𝐴𝐵”〉 = {0, 1} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1507 ∈ wcel 2050 Vcvv 3409 {cpr 4437 dom cdm 5401 ⟶wf 6178 ‘cfv 6182 (class class class)co 6970 0cc0 10329 1c1 10330 2c2 11489 ..^cfzo 12843 ♯chash 13499 Word cword 13666 〈“cs2 14059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10385 ax-resscn 10386 ax-1cn 10387 ax-icn 10388 ax-addcl 10389 ax-addrcl 10390 ax-mulcl 10391 ax-mulrcl 10392 ax-mulcom 10393 ax-addass 10394 ax-mulass 10395 ax-distr 10396 ax-i2m1 10397 ax-1ne0 10398 ax-1rid 10399 ax-rnegex 10400 ax-rrecex 10401 ax-cnre 10402 ax-pre-lttri 10403 ax-pre-lttrn 10404 ax-pre-ltadd 10405 ax-pre-mulgt0 10406 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7495 df-2nd 7496 df-wrecs 7744 df-recs 7806 df-rdg 7844 df-1o 7899 df-oadd 7903 df-er 8083 df-en 8301 df-dom 8302 df-sdom 8303 df-fin 8304 df-card 9156 df-pnf 10470 df-mnf 10471 df-xr 10472 df-ltxr 10473 df-le 10474 df-sub 10666 df-neg 10667 df-nn 11434 df-2 11497 df-n0 11702 df-z 11788 df-uz 12053 df-fz 12703 df-fzo 12844 df-hash 13500 df-word 13667 df-concat 13728 df-s1 13753 df-s2 14066 |
This theorem is referenced by: (None) |
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