MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elovmpt2wrd Structured version   Unicode version

Theorem elovmpt2wrd 12491
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. Note that  ph may depend on  z as well as on  v and  y. (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 12478 . . . . . . . 8  |-  ( v  e.  _V  ->  [_ v  /  x ]_Word  x  = Word  v )
32eqcomd 2390 . . . . . . 7  |-  ( v  e.  _V  -> Word  v  = 
[_ v  /  x ]_Word  x )
43adantr 463 . . . . . 6  |-  ( ( v  e.  _V  /\  y  e.  _V )  -> Word  v  =  [_ v  /  x ]_Word  x )
5 rabeq 3028 . . . . . 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 2411 . . 3  |-  O  =  ( v  e.  _V ,  y  e.  _V  |->  { z  e.  [_ v  /  x ]_Word  x  | 
ph } )
9 csbwrdg 12478 . . . . 5  |-  ( V  e.  _V  ->  [_ V  /  x ]_Word  x  = Word  V )
10 wrdexg 12464 . . . . 5  |-  ( V  e.  _V  -> Word  V  e. 
_V )
119, 10eqeltrd 2470 . . . 4  |-  ( V  e.  _V  ->  [_ V  /  x ]_Word  x  e.  _V )
1211adantr 463 . . 3  |-  ( ( V  e.  _V  /\  Y  e.  _V )  ->  [_ V  /  x ]_Word  x  e.  _V )
138, 12elovmpt2rab1 6421 . 2  |-  ( Z  e.  ( V O Y )  ->  ( V  e.  _V  /\  Y  e.  _V  /\  Z  e. 
[_ V  /  x ]_Word  x ) )
149eleq2d 2452 . . . . 5  |-  ( V  e.  _V  ->  ( Z  e.  [_ V  /  x ]_Word  x  <->  Z  e. Word  V ) )
1514adantr 463 . . . 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 1196 . . . 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 1191 . 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826   {crab 2736   _Vcvv 3034   [_csb 3348  (class class class)co 6196    |-> cmpt2 6198  Word cword 12438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700  df-map 7340  df-pm 7341  df-neg 9721  df-z 10782  df-uz 11002  df-fz 11594  df-fzo 11718  df-word 12446
This theorem is referenced by:  wwlkprop  24806
  Copyright terms: Public domain W3C validator