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Theorem marypha2lem3 7190
Description: Lemma for marypha2 7192. 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 5568 . . . . . . 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 4079 . . . . 5  |-  ( x  e.  A  |->  ( G `
 x ) )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( G `  x ) ) }
53, 4syl6eq 2331 . . . 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 7189 . . . . 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 3207 . . 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 4291 . . 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 1831 . . . . 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 1553 . . . . 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 5539 . . . . . . 7  |-  ( G `
 x )  e. 
_V
16 eleq1 2343 . . . . . . 7  |-  ( y  =  ( G `  x )  ->  (
y  e.  ( F `
 x )  <->  ( G `  x )  e.  ( F `  x ) ) )
1715, 16ceqsalv 2814 . . . . . 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 1553 . . 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 2548 . . 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 1527    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   {csn 3640   U_ciun 3905   {copab 4076    e. cmpt 4077    X. cxp 4687    Fn wfn 5250   ` cfv 5255
This theorem is referenced by:  marypha2  7192
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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