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Mirrors > Home > MPE Home > Th. List > Mathboxes > mrexval | Structured version Visualization version GIF version |
Description: The set of "raw expressions", which are expressions without a typecode, that is, just sequences of constants and variables. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mrexval.c | ⊢ 𝐶 = (mCN‘𝑇) |
mrexval.v | ⊢ 𝑉 = (mVR‘𝑇) |
mrexval.r | ⊢ 𝑅 = (mREx‘𝑇) |
Ref | Expression |
---|---|
mrexval | ⊢ (𝑇 ∈ 𝑊 → 𝑅 = Word (𝐶 ∪ 𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mrexval.r | . 2 ⊢ 𝑅 = (mREx‘𝑇) | |
2 | elex 3504 | . . 3 ⊢ (𝑇 ∈ 𝑊 → 𝑇 ∈ V) | |
3 | fveq2 6919 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (mCN‘𝑡) = (mCN‘𝑇)) | |
4 | mrexval.c | . . . . . . 7 ⊢ 𝐶 = (mCN‘𝑇) | |
5 | 3, 4 | eqtr4di 2792 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mCN‘𝑡) = 𝐶) |
6 | fveq2 6919 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇)) | |
7 | mrexval.v | . . . . . . 7 ⊢ 𝑉 = (mVR‘𝑇) | |
8 | 6, 7 | eqtr4di 2792 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉) |
9 | 5, 8 | uneq12d 4186 | . . . . 5 ⊢ (𝑡 = 𝑇 → ((mCN‘𝑡) ∪ (mVR‘𝑡)) = (𝐶 ∪ 𝑉)) |
10 | wrdeq 14580 | . . . . 5 ⊢ (((mCN‘𝑡) ∪ (mVR‘𝑡)) = (𝐶 ∪ 𝑉) → Word ((mCN‘𝑡) ∪ (mVR‘𝑡)) = Word (𝐶 ∪ 𝑉)) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ (𝑡 = 𝑇 → Word ((mCN‘𝑡) ∪ (mVR‘𝑡)) = Word (𝐶 ∪ 𝑉)) |
12 | df-mrex 35446 | . . . 4 ⊢ mREx = (𝑡 ∈ V ↦ Word ((mCN‘𝑡) ∪ (mVR‘𝑡))) | |
13 | fvex 6932 | . . . . . 6 ⊢ (mCN‘𝑡) ∈ V | |
14 | fvex 6932 | . . . . . 6 ⊢ (mVR‘𝑡) ∈ V | |
15 | 13, 14 | unex 7775 | . . . . 5 ⊢ ((mCN‘𝑡) ∪ (mVR‘𝑡)) ∈ V |
16 | 15 | wrdexi 14570 | . . . 4 ⊢ Word ((mCN‘𝑡) ∪ (mVR‘𝑡)) ∈ V |
17 | 11, 12, 16 | fvmpt3i 7032 | . . 3 ⊢ (𝑇 ∈ V → (mREx‘𝑇) = Word (𝐶 ∪ 𝑉)) |
18 | 2, 17 | syl 17 | . 2 ⊢ (𝑇 ∈ 𝑊 → (mREx‘𝑇) = Word (𝐶 ∪ 𝑉)) |
19 | 1, 18 | eqtrid 2786 | 1 ⊢ (𝑇 ∈ 𝑊 → 𝑅 = Word (𝐶 ∪ 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2103 Vcvv 3482 ∪ cun 3968 ‘cfv 6572 Word cword 14558 mCNcmcn 35420 mVRcmvar 35421 mRExcmrex 35426 |
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-map 8882 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-n0 12550 df-z 12636 df-uz 12900 df-fz 13564 df-fzo 13708 df-hash 14376 df-word 14559 df-mrex 35446 |
This theorem is referenced by: mexval2 35463 mrsubcv 35470 mrsubff 35472 mrsubrn 35473 mrsub0 35476 mrsubccat 35478 elmrsubrn 35480 mrsubco 35481 mrsubvrs 35482 mvhf 35518 msubvrs 35520 |
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