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Theorem marypha2lem4 7958
Description: Lemma for marypha2 7959. 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 7956 . . . . 5  |-  T  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  ( F `  x
) ) }
32imaeq1i 5185 . . . 4  |-  ( T
" X )  =  ( { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( F `  x ) ) }
" X )
4 df-ima 4867 . . . 4  |-  ( {
<. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( F `  x
) ) } " X )  =  ran  ( { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  ( F `  x
) ) }  |`  X )
53, 4eqtri 2458 . . 3  |-  ( T
" X )  =  ran  ( { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( F `  x ) ) }  |`  X )
6 resopab2 5173 . . . . . 6  |-  ( X 
C_  A  ->  ( { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  ( F `  x
) ) }  |`  X )  =  { <. x ,  y >.  |  ( x  e.  X  /\  y  e.  ( F `  x ) ) } )
76adantl 467 . . . . 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 5082 . . . 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 5099 . . . . 5  |-  ran  { <. x ,  y >.  |  ( x  e.  X  /\  y  e.  ( F `  x
) ) }  =  { y  |  E. x ( x  e.  X  /\  y  e.  ( F `  x
) ) }
10 df-rex 2788 . . . . . . . . 9  |-  ( E. x  e.  X  y  e.  ( F `  x )  <->  E. x
( x  e.  X  /\  y  e.  ( F `  x )
) )
1110bicomi 205 . . . . . . . 8  |-  ( E. x ( x  e.  X  /\  y  e.  ( F `  x
) )  <->  E. x  e.  X  y  e.  ( F `  x ) )
1211abbii 2563 . . . . . . 7  |-  { y  |  E. x ( x  e.  X  /\  y  e.  ( F `  x ) ) }  =  { y  |  E. x  e.  X  y  e.  ( F `  x ) }
13 df-iun 4304 . . . . . . 7  |-  U_ x  e.  X  ( F `  x )  =  {
y  |  E. x  e.  X  y  e.  ( F `  x ) }
1412, 13eqtr4i 2461 . . . . . 6  |-  { y  |  E. x ( x  e.  X  /\  y  e.  ( F `  x ) ) }  =  U_ x  e.  X  ( F `  x )
1514a1i 11 . . . . 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 2482 . . . 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 2470 . . 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 2482 . 2  |-  ( ( F  Fn  A  /\  X  C_  A )  -> 
( T " X
)  =  U_ x  e.  X  ( F `  x ) )
19 fnfun 5691 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
2019adantr 466 . . 3  |-  ( ( F  Fn  A  /\  X  C_  A )  ->  Fun  F )
21 funiunfv 6168 . . 3  |-  ( Fun 
F  ->  U_ x  e.  X  ( F `  x )  =  U. ( F " X ) )
2220, 21syl 17 . 2  |-  ( ( F  Fn  A  /\  X  C_  A )  ->  U_ x  e.  X  ( F `  x )  =  U. ( F
" X ) )
2318, 22eqtrd 2470 1  |-  ( ( F  Fn  A  /\  X  C_  A )  -> 
( T " X
)  =  U. ( F " X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437   E.wex 1659    e. wcel 1870   {cab 2414   E.wrex 2783    C_ wss 3442   {csn 4002   U.cuni 4222   U_ciun 4302   {copab 4483    X. cxp 4852   ran crn 4855    |` cres 4856   "cima 4857   Fun wfun 5595    Fn wfn 5596   ` cfv 5601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-fv 5609
This theorem is referenced by:  marypha2  7959
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