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Theorem marypha2lem3 7206
Description: Lemma for marypha2 7208. 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 5584 . . . . . . 7  |-  ( G  Fn  A  <->  G  =  ( x  e.  A  |->  ( G `  x
) ) )
21biimpi 186 . . . . . 6  |-  ( G  Fn  A  ->  G  =  ( x  e.  A  |->  ( G `  x ) ) )
32adantl 452 . . . . 5  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  G  =  ( x  e.  A  |->  ( G `
 x ) ) )
4 df-mpt 4095 . . . . 5  |-  ( x  e.  A  |->  ( G `
 x ) )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( G `  x ) ) }
53, 4syl6eq 2344 . . . 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 7205 . . . . 5  |-  T  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  ( F `  x
) ) }
87a1i 10 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  T  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( F `  x ) ) } )
95, 8sseq12d 3220 . . 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 4307 . . 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 252 . 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 1843 . . . . 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 670 . . . . . 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 1556 . . . . 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 5555 . . . . . . 7  |-  ( G `
 x )  e. 
_V
16 eleq1 2356 . . . . . . 7  |-  ( y  =  ( G `  x )  ->  (
y  e.  ( F `
 x )  <->  ( G `  x )  e.  ( F `  x ) ) )
1715, 16ceqsalv 2827 . . . . . 6  |-  ( A. y ( y  =  ( G `  x
)  ->  y  e.  ( F `  x ) )  <->  ( G `  x )  e.  ( F `  x ) )
1817imbi2i 303 . . . . 5  |-  ( ( x  e.  A  ->  A. y ( y  =  ( G `  x
)  ->  y  e.  ( F `  x ) ) )  <->  ( x  e.  A  ->  ( G `
 x )  e.  ( F `  x
) ) )
1912, 14, 183bitr3i 266 . . . 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 1556 . . 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 2561 . . 3  |-  ( A. x  e.  A  ( G `  x )  e.  ( F `  x
)  <->  A. x ( x  e.  A  ->  ( G `  x )  e.  ( F `  x
) ) )
2220, 21bitr4i 243 . 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 252 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 176    /\ wa 358   A.wal 1530    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   {csn 3653   U_ciun 3921   {copab 4092    e. cmpt 4093    X. cxp 4703    Fn wfn 5266   ` cfv 5271
This theorem is referenced by:  marypha2  7208
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279
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