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Theorem marypha2lem3 7896
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
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 5912 . . . . . . 7  |-  ( G  Fn  A  <->  G  =  ( x  e.  A  |->  ( G `  x
) ) )
21biimpi 194 . . . . . 6  |-  ( G  Fn  A  ->  G  =  ( x  e.  A  |->  ( G `  x ) ) )
32adantl 466 . . . . 5  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  G  =  ( x  e.  A  |->  ( G `
 x ) ) )
4 df-mpt 4507 . . . . 5  |-  ( x  e.  A  |->  ( G `
 x ) )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( G `  x ) ) }
53, 4syl6eq 2524 . . . 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 7895 . . . . 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 3533 . . 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 4774 . . 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 261 . 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 1930 . . . . 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 689 . . . . . 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 1620 . . . . 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 5875 . . . . . . 7  |-  ( G `
 x )  e. 
_V
16 eleq1 2539 . . . . . . 7  |-  ( y  =  ( G `  x )  ->  (
y  e.  ( F `
 x )  <->  ( G `  x )  e.  ( F `  x ) ) )
1715, 16ceqsalv 3141 . . . . . 6  |-  ( A. y ( y  =  ( G `  x
)  ->  y  e.  ( F `  x ) )  <->  ( G `  x )  e.  ( F `  x ) )
1817imbi2i 312 . . . . 5  |-  ( ( x  e.  A  ->  A. y ( y  =  ( G `  x
)  ->  y  e.  ( F `  x ) ) )  <->  ( x  e.  A  ->  ( G `
 x )  e.  ( F `  x
) ) )
1912, 14, 183bitr3i 275 . . . 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 1620 . . 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 2819 . . 3  |-  ( A. x  e.  A  ( G `  x )  e.  ( F `  x
)  <->  A. x ( x  e.  A  ->  ( G `  x )  e.  ( F `  x
) ) )
2220, 21bitr4i 252 . 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 261 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 184    /\ wa 369   A.wal 1377    = wceq 1379    e. wcel 1767   A.wral 2814    C_ wss 3476   {csn 4027   U_ciun 4325   {copab 4504    |-> cmpt 4505    X. cxp 4997    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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  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-iota 5550  df-fun 5589  df-fn 5590  df-fv 5595
This theorem is referenced by:  marypha2  7898
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