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Theorem marypha2lem4 7897
Description: Lemma for marypha2 7898. 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 7895 . . . . 5  |-  T  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  ( F `  x
) ) }
32imaeq1i 5333 . . . 4  |-  ( T
" X )  =  ( { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( F `  x ) ) }
" X )
4 df-ima 5012 . . . 4  |-  ( {
<. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( F `  x
) ) } " X )  =  ran  ( { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  ( F `  x
) ) }  |`  X )
53, 4eqtri 2496 . . 3  |-  ( T
" X )  =  ran  ( { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( F `  x ) ) }  |`  X )
6 resopab2 5321 . . . . . 6  |-  ( X 
C_  A  ->  ( { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  ( F `  x
) ) }  |`  X )  =  { <. x ,  y >.  |  ( x  e.  X  /\  y  e.  ( F `  x ) ) } )
76adantl 466 . . . . 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 5229 . . . 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 5246 . . . . 5  |-  ran  { <. x ,  y >.  |  ( x  e.  X  /\  y  e.  ( F `  x
) ) }  =  { y  |  E. x ( x  e.  X  /\  y  e.  ( F `  x
) ) }
10 df-rex 2820 . . . . . . . . 9  |-  ( E. x  e.  X  y  e.  ( F `  x )  <->  E. x
( x  e.  X  /\  y  e.  ( F `  x )
) )
1110bicomi 202 . . . . . . . 8  |-  ( E. x ( x  e.  X  /\  y  e.  ( F `  x
) )  <->  E. x  e.  X  y  e.  ( F `  x ) )
1211abbii 2601 . . . . . . 7  |-  { y  |  E. x ( x  e.  X  /\  y  e.  ( F `  x ) ) }  =  { y  |  E. x  e.  X  y  e.  ( F `  x ) }
13 df-iun 4327 . . . . . . 7  |-  U_ x  e.  X  ( F `  x )  =  {
y  |  E. x  e.  X  y  e.  ( F `  x ) }
1412, 13eqtr4i 2499 . . . . . 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 2520 . . . 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 2508 . . 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 2520 . 2  |-  ( ( F  Fn  A  /\  X  C_  A )  -> 
( T " X
)  =  U_ x  e.  X  ( F `  x ) )
19 fnfun 5677 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
2019adantr 465 . . 3  |-  ( ( F  Fn  A  /\  X  C_  A )  ->  Fun  F )
21 funiunfv 6147 . . 3  |-  ( Fun 
F  ->  U_ x  e.  X  ( F `  x )  =  U. ( F " X ) )
2220, 21syl 16 . 2  |-  ( ( F  Fn  A  /\  X  C_  A )  ->  U_ x  e.  X  ( F `  x )  =  U. ( F
" X ) )
2318, 22eqtrd 2508 1  |-  ( ( F  Fn  A  /\  X  C_  A )  -> 
( T " X
)  =  U. ( F " X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   {cab 2452   E.wrex 2815    C_ wss 3476   {csn 4027   U.cuni 4245   U_ciun 4325   {copab 4504    X. cxp 4997   ran crn 5000    |` cres 5001   "cima 5002   Fun wfun 5581    Fn wfn 5582   ` cfv 5587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-fv 5595
This theorem is referenced by:  marypha2  7898
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