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Theorem elovmpt2wrd 30405
Description: Implications for the value of an operation defined by the maps-to notation with a class abstration of words as a result having an element. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
Hypothesis
Ref Expression
elovmpt2wrd.o  |-  O  =  ( v  e.  _V ,  y  e.  _V  |->  { z  e. Word  v  |  ph } )
Assertion
Ref Expression
elovmpt2wrd  |-  ( Z  e.  ( V O Y )  ->  ( V  e.  _V  /\  Y  e.  _V  /\  Z  e. Word  V ) )
Distinct variable groups:    v, V, y, z    v, Y, y, z    z, Z
Allowed substitution hints:    ph( y, z, v)    O( y, z, v)    Z( y, v)

Proof of Theorem elovmpt2wrd
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elovmpt2wrd.o . . . 4  |-  O  =  ( v  e.  _V ,  y  e.  _V  |->  { z  e. Word  v  |  ph } )
2 csbwrdg 12376 . . . . . . . 8  |-  ( v  e.  _V  ->  [_ v  /  x ]_Word  x  = Word  v )
32eqcomd 2462 . . . . . . 7  |-  ( v  e.  _V  -> Word  v  = 
[_ v  /  x ]_Word  x )
43adantr 465 . . . . . 6  |-  ( ( v  e.  _V  /\  y  e.  _V )  -> Word  v  =  [_ v  /  x ]_Word  x )
5 rabeq 3072 . . . . . 6  |-  (Word  v  =  [_ v  /  x ]_Word  x  ->  { z  e. Word  v  |  ph }  =  { z  e.  [_ v  /  x ]_Word  x  | 
ph } )
64, 5syl 16 . . . . 5  |-  ( ( v  e.  _V  /\  y  e.  _V )  ->  { z  e. Word  v  |  ph }  =  {
z  e.  [_ v  /  x ]_Word  x  |  ph } )
76mpt2eq3ia 6261 . . . 4  |-  ( v  e.  _V ,  y  e.  _V  |->  { z  e. Word  v  |  ph } )  =  ( v  e.  _V , 
y  e.  _V  |->  { z  e.  [_ v  /  x ]_Word  x  |  ph } )
81, 7eqtri 2483 . . 3  |-  O  =  ( v  e.  _V ,  y  e.  _V  |->  { z  e.  [_ v  /  x ]_Word  x  | 
ph } )
9 csbwrdg 12376 . . . . 5  |-  ( V  e.  _V  ->  [_ V  /  x ]_Word  x  = Word  V )
10 wrdexg 12363 . . . . 5  |-  ( V  e.  _V  -> Word  V  e. 
_V )
119, 10eqeltrd 2542 . . . 4  |-  ( V  e.  _V  ->  [_ V  /  x ]_Word  x  e.  _V )
1211adantr 465 . . 3  |-  ( ( V  e.  _V  /\  Y  e.  _V )  ->  [_ V  /  x ]_Word  x  e.  _V )
138, 12elovmpt2rab1 30308 . 2  |-  ( Z  e.  ( V O Y )  ->  ( V  e.  _V  /\  Y  e.  _V  /\  Z  e. 
[_ V  /  x ]_Word  x ) )
149eleq2d 2524 . . . . 5  |-  ( V  e.  _V  ->  ( Z  e.  [_ V  /  x ]_Word  x  <->  Z  e. Word  V ) )
1514adantr 465 . . . 4  |-  ( ( V  e.  _V  /\  Y  e.  _V )  ->  ( Z  e.  [_ V  /  x ]_Word  x  <->  Z  e. Word  V ) )
16 id 22 . . . . 5  |-  ( ( V  e.  _V  /\  Y  e.  _V  /\  Z  e. Word  V )  ->  ( V  e.  _V  /\  Y  e.  _V  /\  Z  e. Word  V ) )
17163expia 1190 . . . 4  |-  ( ( V  e.  _V  /\  Y  e.  _V )  ->  ( Z  e. Word  V  ->  ( V  e.  _V  /\  Y  e.  _V  /\  Z  e. Word  V )
) )
1815, 17sylbid 215 . . 3  |-  ( ( V  e.  _V  /\  Y  e.  _V )  ->  ( Z  e.  [_ V  /  x ]_Word  x  -> 
( V  e.  _V  /\  Y  e.  _V  /\  Z  e. Word  V )
) )
19183impia 1185 . 2  |-  ( ( V  e.  _V  /\  Y  e.  _V  /\  Z  e.  [_ V  /  x ]_Word  x )  ->  ( V  e.  _V  /\  Y  e.  _V  /\  Z  e. Word  V ) )
2013, 19syl 16 1  |-  ( Z  e.  ( V O Y )  ->  ( V  e.  _V  /\  Y  e.  _V  /\  Z  e. Word  V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   {crab 2803   _Vcvv 3078   [_csb 3396  (class class class)co 6201    |-> cmpt2 6203  Word cword 12340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689  df-map 7327  df-pm 7328  df-neg 9710  df-z 10759  df-uz 10974  df-fz 11556  df-fzo 11667  df-word 12348
This theorem is referenced by:  wwlkprop  30468
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