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Theorem marypha2lem3 7957
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
marypha2lem3  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( G  C_  T  <->  A. x  e.  A  ( G `  x )  e.  ( F `  x ) ) )
Distinct variable groups:    x, A    x, F    x, G
Allowed substitution hint:    T( x)

Proof of Theorem marypha2lem3
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dffn5 5926 . . . . . . 7  |-  ( G  Fn  A  <->  G  =  ( x  e.  A  |->  ( G `  x
) ) )
21biimpi 197 . . . . . 6  |-  ( G  Fn  A  ->  G  =  ( x  e.  A  |->  ( G `  x ) ) )
32adantl 467 . . . . 5  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  G  =  ( x  e.  A  |->  ( G `
 x ) ) )
4 df-mpt 4486 . . . . 5  |-  ( x  e.  A  |->  ( G `
 x ) )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( G `  x ) ) }
53, 4syl6eq 2486 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  G  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( G `  x ) ) } )
6 marypha2lem.t . . . . . 6  |-  T  = 
U_ x  e.  A  ( { x }  X.  ( F `  x ) )
76marypha2lem2 7956 . . . . 5  |-  T  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  ( F `  x
) ) }
87a1i 11 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  T  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( F `  x ) ) } )
95, 8sseq12d 3499 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( G  C_  T  <->  {
<. x ,  y >.  |  ( x  e.  A  /\  y  =  ( G `  x
) ) }  C_  {
<. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( F `  x
) ) } ) )
10 ssopab2b 4748 . . 3  |-  ( {
<. x ,  y >.  |  ( x  e.  A  /\  y  =  ( G `  x
) ) }  C_  {
<. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( F `  x
) ) }  <->  A. x A. y ( ( x  e.  A  /\  y  =  ( G `  x ) )  -> 
( x  e.  A  /\  y  e.  ( F `  x )
) ) )
119, 10syl6bb 264 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( G  C_  T  <->  A. x A. y ( ( x  e.  A  /\  y  =  ( G `  x )
)  ->  ( x  e.  A  /\  y  e.  ( F `  x
) ) ) ) )
12 19.21v 1778 . . . . 5  |-  ( A. y ( x  e.  A  ->  ( y  =  ( G `  x )  ->  y  e.  ( F `  x
) ) )  <->  ( x  e.  A  ->  A. y
( y  =  ( G `  x )  ->  y  e.  ( F `  x ) ) ) )
13 imdistan 693 . . . . . 6  |-  ( ( x  e.  A  -> 
( y  =  ( G `  x )  ->  y  e.  ( F `  x ) ) )  <->  ( (
x  e.  A  /\  y  =  ( G `  x ) )  -> 
( x  e.  A  /\  y  e.  ( F `  x )
) ) )
1413albii 1687 . . . . 5  |-  ( A. y ( x  e.  A  ->  ( y  =  ( G `  x )  ->  y  e.  ( F `  x
) ) )  <->  A. y
( ( x  e.  A  /\  y  =  ( G `  x
) )  ->  (
x  e.  A  /\  y  e.  ( F `  x ) ) ) )
15 fvex 5891 . . . . . . 7  |-  ( G `
 x )  e. 
_V
16 eleq1 2501 . . . . . . 7  |-  ( y  =  ( G `  x )  ->  (
y  e.  ( F `
 x )  <->  ( G `  x )  e.  ( F `  x ) ) )
1715, 16ceqsalv 3115 . . . . . 6  |-  ( A. y ( y  =  ( G `  x
)  ->  y  e.  ( F `  x ) )  <->  ( G `  x )  e.  ( F `  x ) )
1817imbi2i 313 . . . . 5  |-  ( ( x  e.  A  ->  A. y ( y  =  ( G `  x
)  ->  y  e.  ( F `  x ) ) )  <->  ( x  e.  A  ->  ( G `
 x )  e.  ( F `  x
) ) )
1912, 14, 183bitr3i 278 . . . 4  |-  ( A. y ( ( x  e.  A  /\  y  =  ( G `  x ) )  -> 
( x  e.  A  /\  y  e.  ( F `  x )
) )  <->  ( x  e.  A  ->  ( G `
 x )  e.  ( F `  x
) ) )
2019albii 1687 . . 3  |-  ( A. x A. y ( ( x  e.  A  /\  y  =  ( G `  x ) )  -> 
( x  e.  A  /\  y  e.  ( F `  x )
) )  <->  A. x
( x  e.  A  ->  ( G `  x
)  e.  ( F `
 x ) ) )
21 df-ral 2787 . . 3  |-  ( A. x  e.  A  ( G `  x )  e.  ( F `  x
)  <->  A. x ( x  e.  A  ->  ( G `  x )  e.  ( F `  x
) ) )
2220, 21bitr4i 255 . 2  |-  ( A. x A. y ( ( x  e.  A  /\  y  =  ( G `  x ) )  -> 
( x  e.  A  /\  y  e.  ( F `  x )
) )  <->  A. x  e.  A  ( G `  x )  e.  ( F `  x ) )
2311, 22syl6bb 264 1  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( G  C_  T  <->  A. x  e.  A  ( G `  x )  e.  ( F `  x ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435    = wceq 1437    e. wcel 1870   A.wral 2782    C_ wss 3442   {csn 4002   U_ciun 4302   {copab 4483    |-> cmpt 4484    X. cxp 4852    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-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  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-iota 5565  df-fun 5603  df-fn 5604  df-fv 5609
This theorem is referenced by:  marypha2  7959
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