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Mirrors > Home > MPE Home > Th. List > Mathboxes > mvhf | Structured version Visualization version GIF version |
Description: The function mapping variables to variable expressions is a function. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mvhf.v | ⊢ 𝑉 = (mVR‘𝑇) |
mvhf.e | ⊢ 𝐸 = (mEx‘𝑇) |
mvhf.h | ⊢ 𝐻 = (mVH‘𝑇) |
Ref | Expression |
---|---|
mvhf | ⊢ (𝑇 ∈ mFS → 𝐻:𝑉⟶𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mvhf.v | . . . . . 6 ⊢ 𝑉 = (mVR‘𝑇) | |
2 | eqid 2740 | . . . . . 6 ⊢ (mTC‘𝑇) = (mTC‘𝑇) | |
3 | eqid 2740 | . . . . . 6 ⊢ (mType‘𝑇) = (mType‘𝑇) | |
4 | 1, 2, 3 | mtyf2 33509 | . . . . 5 ⊢ (𝑇 ∈ mFS → (mType‘𝑇):𝑉⟶(mTC‘𝑇)) |
5 | 4 | ffvelrnda 6958 | . . . 4 ⊢ ((𝑇 ∈ mFS ∧ 𝑣 ∈ 𝑉) → ((mType‘𝑇)‘𝑣) ∈ (mTC‘𝑇)) |
6 | elun2 4116 | . . . . . . 7 ⊢ (𝑣 ∈ 𝑉 → 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) | |
7 | 6 | adantl 482 | . . . . . 6 ⊢ ((𝑇 ∈ mFS ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) |
8 | 7 | s1cld 14306 | . . . . 5 ⊢ ((𝑇 ∈ mFS ∧ 𝑣 ∈ 𝑉) → 〈“𝑣”〉 ∈ Word ((mCN‘𝑇) ∪ 𝑉)) |
9 | eqid 2740 | . . . . . . 7 ⊢ (mCN‘𝑇) = (mCN‘𝑇) | |
10 | eqid 2740 | . . . . . . 7 ⊢ (mREx‘𝑇) = (mREx‘𝑇) | |
11 | 9, 1, 10 | mrexval 33459 | . . . . . 6 ⊢ (𝑇 ∈ mFS → (mREx‘𝑇) = Word ((mCN‘𝑇) ∪ 𝑉)) |
12 | 11 | adantr 481 | . . . . 5 ⊢ ((𝑇 ∈ mFS ∧ 𝑣 ∈ 𝑉) → (mREx‘𝑇) = Word ((mCN‘𝑇) ∪ 𝑉)) |
13 | 8, 12 | eleqtrrd 2844 | . . . 4 ⊢ ((𝑇 ∈ mFS ∧ 𝑣 ∈ 𝑉) → 〈“𝑣”〉 ∈ (mREx‘𝑇)) |
14 | opelxpi 5627 | . . . 4 ⊢ ((((mType‘𝑇)‘𝑣) ∈ (mTC‘𝑇) ∧ 〈“𝑣”〉 ∈ (mREx‘𝑇)) → 〈((mType‘𝑇)‘𝑣), 〈“𝑣”〉〉 ∈ ((mTC‘𝑇) × (mREx‘𝑇))) | |
15 | 5, 13, 14 | syl2anc 584 | . . 3 ⊢ ((𝑇 ∈ mFS ∧ 𝑣 ∈ 𝑉) → 〈((mType‘𝑇)‘𝑣), 〈“𝑣”〉〉 ∈ ((mTC‘𝑇) × (mREx‘𝑇))) |
16 | mvhf.e | . . . 4 ⊢ 𝐸 = (mEx‘𝑇) | |
17 | 2, 16, 10 | mexval 33460 | . . 3 ⊢ 𝐸 = ((mTC‘𝑇) × (mREx‘𝑇)) |
18 | 15, 17 | eleqtrrdi 2852 | . 2 ⊢ ((𝑇 ∈ mFS ∧ 𝑣 ∈ 𝑉) → 〈((mType‘𝑇)‘𝑣), 〈“𝑣”〉〉 ∈ 𝐸) |
19 | mvhf.h | . . 3 ⊢ 𝐻 = (mVH‘𝑇) | |
20 | 1, 3, 19 | mvhfval 33491 | . 2 ⊢ 𝐻 = (𝑣 ∈ 𝑉 ↦ 〈((mType‘𝑇)‘𝑣), 〈“𝑣”〉〉) |
21 | 18, 20 | fmptd 6985 | 1 ⊢ (𝑇 ∈ mFS → 𝐻:𝑉⟶𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ∪ cun 3890 〈cop 4573 × cxp 5588 ⟶wf 6428 ‘cfv 6432 Word cword 14215 〈“cs1 14298 mCNcmcn 33418 mVRcmvar 33419 mTypecmty 33420 mTCcmtc 33422 mRExcmrex 33424 mExcmex 33425 mVHcmvh 33430 mFScmfs 33434 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-map 8600 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12582 df-fz 13239 df-fzo 13382 df-hash 14043 df-word 14216 df-s1 14299 df-mrex 33444 df-mex 33445 df-mvh 33450 df-mfs 33454 |
This theorem is referenced by: mvhf1 33517 msubvrs 33518 mclsssvlem 33520 vhmcls 33524 mclsax 33527 mclsind 33528 mclsppslem 33541 mclspps 33542 |
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