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Theorem marypha2lem4 7207
Description: Lemma for marypha2 7208. 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 7205 . . . . 5  |-  T  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  ( F `  x
) ) }
32imaeq1i 5025 . . . 4  |-  ( T
" X )  =  ( { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( F `  x ) ) }
" X )
4 df-ima 4718 . . . 4  |-  ( {
<. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( F `  x
) ) } " X )  =  ran  ( { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  ( F `  x
) ) }  |`  X )
53, 4eqtri 2316 . . 3  |-  ( T
" X )  =  ran  ( { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( F `  x ) ) }  |`  X )
6 resopab2 5015 . . . . . 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 4922 . . . 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 4940 . . . . 5  |-  ran  { <. x ,  y >.  |  ( x  e.  X  /\  y  e.  ( F `  x
) ) }  =  { y  |  E. x ( x  e.  X  /\  y  e.  ( F `  x
) ) }
10 df-rex 2562 . . . . . . . . 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 2408 . . . . . . 7  |-  { y  |  E. x ( x  e.  X  /\  y  e.  ( F `  x ) ) }  =  { y  |  E. x  e.  X  y  e.  ( F `  x ) }
13 df-iun 3923 . . . . . . 7  |-  U_ x  e.  X  ( F `  x )  =  {
y  |  E. x  e.  X  y  e.  ( F `  x ) }
1412, 13eqtr4i 2319 . . . . . 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 2340 . . . 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 2328 . . 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 2340 . 2  |-  ( ( F  Fn  A  /\  X  C_  A )  -> 
( T " X
)  =  U_ x  e.  X  ( F `  x ) )
19 fnfun 5357 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
2019adantr 451 . . 3  |-  ( ( F  Fn  A  /\  X  C_  A )  ->  Fun  F )
21 funiunfv 5790 . . 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 2328 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 1531    = wceq 1632    e. wcel 1696   {cab 2282   E.wrex 2557    C_ wss 3165   {csn 3653   U.cuni 3843   U_ciun 3921   {copab 4092    X. cxp 4703   ran crn 4706    |` cres 4707   "cima 4708   Fun wfun 5265    Fn wfn 5266   ` cfv 5271
This theorem is referenced by:  marypha2  7208
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279
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