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Theorem marypha2lem3 7444
Description: Lemma for marypha2 7446. 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 5774 . . . . . . 7  |-  ( G  Fn  A  <->  G  =  ( x  e.  A  |->  ( G `  x
) ) )
21biimpi 188 . . . . . 6  |-  ( G  Fn  A  ->  G  =  ( x  e.  A  |->  ( G `  x ) ) )
32adantl 454 . . . . 5  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  G  =  ( x  e.  A  |->  ( G `
 x ) ) )
4 df-mpt 4270 . . . . 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 7443 . . . . 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 3379 . . 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 4483 . . 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 254 . 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 1914 . . . . 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 672 . . . . . 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 1576 . . . . 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 5744 . . . . . . 7  |-  ( G `
 x )  e. 
_V
16 eleq1 2498 . . . . . . 7  |-  ( y  =  ( G `  x )  ->  (
y  e.  ( F `
 x )  <->  ( G `  x )  e.  ( F `  x ) ) )
1715, 16ceqsalv 2984 . . . . . 6  |-  ( A. y ( y  =  ( G `  x
)  ->  y  e.  ( F `  x ) )  <->  ( G `  x )  e.  ( F `  x ) )
1817imbi2i 305 . . . . 5  |-  ( ( x  e.  A  ->  A. y ( y  =  ( G `  x
)  ->  y  e.  ( F `  x ) ) )  <->  ( x  e.  A  ->  ( G `
 x )  e.  ( F `  x
) ) )
1912, 14, 183bitr3i 268 . . . 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 1576 . . 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 2712 . . 3  |-  ( A. x  e.  A  ( G `  x )  e.  ( F `  x
)  <->  A. x ( x  e.  A  ->  ( G `  x )  e.  ( F `  x
) ) )
2220, 21bitr4i 245 . 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 254 1  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( G  C_  T  <->  A. x  e.  A  ( G `  x )  e.  ( F `  x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   A.wal 1550    = wceq 1653    e. wcel 1726   A.wral 2707    C_ wss 3322   {csn 3816   U_ciun 4095   {copab 4267    e. cmpt 4268    X. cxp 4878    Fn wfn 5451   ` cfv 5456
This theorem is referenced by:  marypha2  7446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fn 5459  df-fv 5464
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