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Theorem mvhf1 29186
Description: The function mapping variables to variable expressions is one-to-one. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mvhf.v  |-  V  =  (mVR `  T )
mvhf.e  |-  E  =  (mEx `  T )
mvhf.h  |-  H  =  (mVH `  T )
Assertion
Ref Expression
mvhf1  |-  ( T  e. mFS  ->  H : V -1-1-> E )

Proof of Theorem mvhf1
Dummy variables  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mvhf.v . . 3  |-  V  =  (mVR `  T )
2 mvhf.e . . 3  |-  E  =  (mEx `  T )
3 mvhf.h . . 3  |-  H  =  (mVH `  T )
41, 2, 3mvhf 29185 . 2  |-  ( T  e. mFS  ->  H : V --> E )
5 eqid 2454 . . . . . . 7  |-  (mType `  T )  =  (mType `  T )
61, 5, 3mvhval 29161 . . . . . 6  |-  ( v  e.  V  ->  ( H `  v )  =  <. ( (mType `  T ) `  v
) ,  <" v "> >. )
71, 5, 3mvhval 29161 . . . . . 6  |-  ( w  e.  V  ->  ( H `  w )  =  <. ( (mType `  T ) `  w
) ,  <" w "> >. )
86, 7eqeqan12d 2477 . . . . 5  |-  ( ( v  e.  V  /\  w  e.  V )  ->  ( ( H `  v )  =  ( H `  w )  <->  <. ( (mType `  T
) `  v ) ,  <" v "> >.  =  <. (
(mType `  T ) `  w ) ,  <" w "> >. )
)
98adantl 464 . . . 4  |-  ( ( T  e. mFS  /\  (
v  e.  V  /\  w  e.  V )
)  ->  ( ( H `  v )  =  ( H `  w )  <->  <. ( (mType `  T ) `  v
) ,  <" v "> >.  =  <. (
(mType `  T ) `  w ) ,  <" w "> >. )
)
10 fvex 5858 . . . . . . 7  |-  ( (mType `  T ) `  v
)  e.  _V
11 s1cli 12608 . . . . . . . 8  |-  <" v ">  e. Word  _V
1211elexi 3116 . . . . . . 7  |-  <" v ">  e.  _V
1310, 12opth 4711 . . . . . 6  |-  ( <.
( (mType `  T
) `  v ) ,  <" v "> >.  =  <. (
(mType `  T ) `  w ) ,  <" w "> >.  <->  ( (
(mType `  T ) `  v )  =  ( (mType `  T ) `  w )  /\  <" v ">  =  <" w "> ) )
1413simprbi 462 . . . . 5  |-  ( <.
( (mType `  T
) `  v ) ,  <" v "> >.  =  <. (
(mType `  T ) `  w ) ,  <" w "> >.  ->  <" v ">  =  <" w "> )
15 s111 12615 . . . . . 6  |-  ( ( v  e.  V  /\  w  e.  V )  ->  ( <" v ">  =  <" w ">  <->  v  =  w ) )
1615adantl 464 . . . . 5  |-  ( ( T  e. mFS  /\  (
v  e.  V  /\  w  e.  V )
)  ->  ( <" v ">  =  <" w ">  <->  v  =  w ) )
1714, 16syl5ib 219 . . . 4  |-  ( ( T  e. mFS  /\  (
v  e.  V  /\  w  e.  V )
)  ->  ( <. ( (mType `  T ) `  v ) ,  <" v "> >.  =  <. ( (mType `  T ) `  w ) ,  <" w "> >.  ->  v  =  w ) )
189, 17sylbid 215 . . 3  |-  ( ( T  e. mFS  /\  (
v  e.  V  /\  w  e.  V )
)  ->  ( ( H `  v )  =  ( H `  w )  ->  v  =  w ) )
1918ralrimivva 2875 . 2  |-  ( T  e. mFS  ->  A. v  e.  V  A. w  e.  V  ( ( H `  v )  =  ( H `  w )  ->  v  =  w ) )
20 dff13 6141 . 2  |-  ( H : V -1-1-> E  <->  ( H : V --> E  /\  A. v  e.  V  A. w  e.  V  (
( H `  v
)  =  ( H `
 w )  -> 
v  =  w ) ) )
214, 19, 20sylanbrc 662 1  |-  ( T  e. mFS  ->  H : V -1-1-> E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   _Vcvv 3106   <.cop 4022   -->wf 5566   -1-1->wf1 5567   ` cfv 5570  Word cword 12521   <"cs1 12524  mVRcmvar 29088  mTypecmty 29089  mExcmex 29094  mVHcmvh 29099  mFScmfs 29103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-hash 12391  df-word 12529  df-s1 12532  df-mrex 29113  df-mex 29114  df-mvh 29119  df-mfs 29123
This theorem is referenced by: (None)
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