MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  marypha2lem4 Unicode version

Theorem marypha2lem4 7191
Description: Lemma for marypha2 7192. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
Hypothesis
Ref Expression
marypha2lem.t  |-  T  = 
U_ x  e.  A  ( { x }  X.  ( F `  x ) )
Assertion
Ref Expression
marypha2lem4  |-  ( ( F  Fn  A  /\  X  C_  A )  -> 
( T " X
)  =  U. ( F " X ) )
Distinct variable groups:    x, A    x, F    x, X
Allowed substitution hint:    T( x)

Proof of Theorem marypha2lem4
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 marypha2lem.t . . . . . 6  |-  T  = 
U_ x  e.  A  ( { x }  X.  ( F `  x ) )
21marypha2lem2 7189 . . . . 5  |-  T  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  ( F `  x
) ) }
32imaeq1i 5009 . . . 4  |-  ( T
" X )  =  ( { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( F `  x ) ) }
" X )
4 df-ima 4702 . . . 4  |-  ( {
<. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( F `  x
) ) } " X )  =  ran  ( { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  ( F `  x
) ) }  |`  X )
53, 4eqtri 2303 . . 3  |-  ( T
" X )  =  ran  ( { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( F `  x ) ) }  |`  X )
6 resopab2 4999 . . . . . 6  |-  ( X 
C_  A  ->  ( { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  ( F `  x
) ) }  |`  X )  =  { <. x ,  y >.  |  ( x  e.  X  /\  y  e.  ( F `  x ) ) } )
76adantl 452 . . . . 5  |-  ( ( F  Fn  A  /\  X  C_  A )  -> 
( { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( F `  x ) ) }  |`  X )  =  { <. x ,  y >.  |  ( x  e.  X  /\  y  e.  ( F `  x
) ) } )
87rneqd 4906 . . . 4  |-  ( ( F  Fn  A  /\  X  C_  A )  ->  ran  ( { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( F `  x ) ) }  |`  X )  =  ran  {
<. x ,  y >.  |  ( x  e.  X  /\  y  e.  ( F `  x
) ) } )
9 rnopab 4924 . . . . 5  |-  ran  { <. x ,  y >.  |  ( x  e.  X  /\  y  e.  ( F `  x
) ) }  =  { y  |  E. x ( x  e.  X  /\  y  e.  ( F `  x
) ) }
10 df-rex 2549 . . . . . . . . 9  |-  ( E. x  e.  X  y  e.  ( F `  x )  <->  E. x
( x  e.  X  /\  y  e.  ( F `  x )
) )
1110bicomi 193 . . . . . . . 8  |-  ( E. x ( x  e.  X  /\  y  e.  ( F `  x
) )  <->  E. x  e.  X  y  e.  ( F `  x ) )
1211abbii 2395 . . . . . . 7  |-  { y  |  E. x ( x  e.  X  /\  y  e.  ( F `  x ) ) }  =  { y  |  E. x  e.  X  y  e.  ( F `  x ) }
13 df-iun 3907 . . . . . . 7  |-  U_ x  e.  X  ( F `  x )  =  {
y  |  E. x  e.  X  y  e.  ( F `  x ) }
1412, 13eqtr4i 2306 . . . . . 6  |-  { y  |  E. x ( x  e.  X  /\  y  e.  ( F `  x ) ) }  =  U_ x  e.  X  ( F `  x )
1514a1i 10 . . . . 5  |-  ( ( F  Fn  A  /\  X  C_  A )  ->  { y  |  E. x ( x  e.  X  /\  y  e.  ( F `  x
) ) }  =  U_ x  e.  X  ( F `  x ) )
169, 15syl5eq 2327 . . . 4  |-  ( ( F  Fn  A  /\  X  C_  A )  ->  ran  { <. x ,  y
>.  |  ( x  e.  X  /\  y  e.  ( F `  x
) ) }  =  U_ x  e.  X  ( F `  x ) )
178, 16eqtrd 2315 . . 3  |-  ( ( F  Fn  A  /\  X  C_  A )  ->  ran  ( { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( F `  x ) ) }  |`  X )  =  U_ x  e.  X  ( F `  x )
)
185, 17syl5eq 2327 . 2  |-  ( ( F  Fn  A  /\  X  C_  A )  -> 
( T " X
)  =  U_ x  e.  X  ( F `  x ) )
19 fnfun 5341 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
2019adantr 451 . . 3  |-  ( ( F  Fn  A  /\  X  C_  A )  ->  Fun  F )
21 funiunfv 5774 . . 3  |-  ( Fun 
F  ->  U_ x  e.  X  ( F `  x )  =  U. ( F " X ) )
2220, 21syl 15 . 2  |-  ( ( F  Fn  A  /\  X  C_  A )  ->  U_ x  e.  X  ( F `  x )  =  U. ( F
" X ) )
2318, 22eqtrd 2315 1  |-  ( ( F  Fn  A  /\  X  C_  A )  -> 
( T " X
)  =  U. ( F " X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544    C_ wss 3152   {csn 3640   U.cuni 3827   U_ciun 3905   {copab 4076    X. cxp 4687   ran crn 4690    |` cres 4691   "cima 4692   Fun wfun 5249    Fn wfn 5250   ` cfv 5255
This theorem is referenced by:  marypha2  7192
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
  Copyright terms: Public domain W3C validator