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Theorem marypha2lem4 7435
Description: Lemma for marypha2 7436. 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 7433 . . . . 5  |-  T  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  ( F `  x
) ) }
32imaeq1i 5192 . . . 4  |-  ( T
" X )  =  ( { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( F `  x ) ) }
" X )
4 df-ima 4883 . . . 4  |-  ( {
<. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( F `  x
) ) } " X )  =  ran  ( { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  ( F `  x
) ) }  |`  X )
53, 4eqtri 2455 . . 3  |-  ( T
" X )  =  ran  ( { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( F `  x ) ) }  |`  X )
6 resopab2 5182 . . . . . 6  |-  ( X 
C_  A  ->  ( { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  ( F `  x
) ) }  |`  X )  =  { <. x ,  y >.  |  ( x  e.  X  /\  y  e.  ( F `  x ) ) } )
76adantl 453 . . . . 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 5089 . . . 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 5107 . . . . 5  |-  ran  { <. x ,  y >.  |  ( x  e.  X  /\  y  e.  ( F `  x
) ) }  =  { y  |  E. x ( x  e.  X  /\  y  e.  ( F `  x
) ) }
10 df-rex 2703 . . . . . . . . 9  |-  ( E. x  e.  X  y  e.  ( F `  x )  <->  E. x
( x  e.  X  /\  y  e.  ( F `  x )
) )
1110bicomi 194 . . . . . . . 8  |-  ( E. x ( x  e.  X  /\  y  e.  ( F `  x
) )  <->  E. x  e.  X  y  e.  ( F `  x ) )
1211abbii 2547 . . . . . . 7  |-  { y  |  E. x ( x  e.  X  /\  y  e.  ( F `  x ) ) }  =  { y  |  E. x  e.  X  y  e.  ( F `  x ) }
13 df-iun 4087 . . . . . . 7  |-  U_ x  e.  X  ( F `  x )  =  {
y  |  E. x  e.  X  y  e.  ( F `  x ) }
1412, 13eqtr4i 2458 . . . . . 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 2479 . . . 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 2467 . . 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 2479 . 2  |-  ( ( F  Fn  A  /\  X  C_  A )  -> 
( T " X
)  =  U_ x  e.  X  ( F `  x ) )
19 fnfun 5534 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
2019adantr 452 . . 3  |-  ( ( F  Fn  A  /\  X  C_  A )  ->  Fun  F )
21 funiunfv 5987 . . 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 2467 1  |-  ( ( F  Fn  A  /\  X  C_  A )  -> 
( T " X
)  =  U. ( F " X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2421   E.wrex 2698    C_ wss 3312   {csn 3806   U.cuni 4007   U_ciun 4085   {copab 4257    X. cxp 4868   ran crn 4871    |` cres 4872   "cima 4873   Fun wfun 5440    Fn wfn 5441   ` cfv 5446
This theorem is referenced by:  marypha2  7436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454
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