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Theorem cvmlift2lem4 23837
Description: Lemma for cvmlift2 23847. (Contributed by Mario Carneiro, 1-Jun-2015.)
Hypotheses
Ref Expression
cvmlift2.b  |-  B  = 
U. C
cvmlift2.f  |-  ( ph  ->  F  e.  ( C CovMap  J ) )
cvmlift2.g  |-  ( ph  ->  G  e.  ( ( II  tX  II )  Cn  J ) )
cvmlift2.p  |-  ( ph  ->  P  e.  B )
cvmlift2.i  |-  ( ph  ->  ( F `  P
)  =  ( 0 G 0 ) )
cvmlift2.h  |-  H  =  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( z G 0 ) )  /\  ( f `
 0 )  =  P ) )
cvmlift2.k  |-  K  =  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( ( iota_ f  e.  ( II  Cn  C
) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 ) 
|->  ( x G z ) )  /\  (
f `  0 )  =  ( H `  x ) ) ) `
 y ) )
Assertion
Ref Expression
cvmlift2lem4  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  e.  ( 0 [,] 1 ) )  ->  ( X K Y )  =  ( ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( X G z ) )  /\  ( f `
 0 )  =  ( H `  X
) ) ) `  Y ) )
Distinct variable groups:    x, f,
y, z, F    ph, f, x, y, z    f, J, x, y, z    f, G, x, y, z    f, H, x, y, z    f, X, x, y, z    C, f, x, y, z    P, f, x, y, z    x, B, y, z    f, Y, x, y, z    f, K, x, y, z
Allowed substitution hint:    B( f)

Proof of Theorem cvmlift2lem4
StepHypRef Expression
1 oveq1 5865 . . . . . . 7  |-  ( x  =  X  ->  (
x G z )  =  ( X G z ) )
21mpteq2dv 4107 . . . . . 6  |-  ( x  =  X  ->  (
z  e.  ( 0 [,] 1 )  |->  ( x G z ) )  =  ( z  e.  ( 0 [,] 1 )  |->  ( X G z ) ) )
32eqeq2d 2294 . . . . 5  |-  ( x  =  X  ->  (
( F  o.  f
)  =  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) )  <-> 
( F  o.  f
)  =  ( z  e.  ( 0 [,] 1 )  |->  ( X G z ) ) ) )
4 fveq2 5525 . . . . . 6  |-  ( x  =  X  ->  ( H `  x )  =  ( H `  X ) )
54eqeq2d 2294 . . . . 5  |-  ( x  =  X  ->  (
( f `  0
)  =  ( H `
 x )  <->  ( f `  0 )  =  ( H `  X
) ) )
63, 5anbi12d 691 . . . 4  |-  ( x  =  X  ->  (
( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) )  /\  ( f `
 0 )  =  ( H `  x
) )  <->  ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1
)  |->  ( X G z ) )  /\  ( f `  0
)  =  ( H `
 X ) ) ) )
76riotabidv 6306 . . 3  |-  ( x  =  X  ->  ( iota_ f  e.  ( II 
Cn  C ) ( ( F  o.  f
)  =  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) )  /\  ( f ` 
0 )  =  ( H `  x ) ) )  =  (
iota_ f  e.  (
II  Cn  C )
( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( X G z ) )  /\  ( f `
 0 )  =  ( H `  X
) ) ) )
87fveq1d 5527 . 2  |-  ( x  =  X  ->  (
( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) )  /\  ( f `
 0 )  =  ( H `  x
) ) ) `  y )  =  ( ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( X G z ) )  /\  ( f `
 0 )  =  ( H `  X
) ) ) `  y ) )
9 fveq2 5525 . 2  |-  ( y  =  Y  ->  (
( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( X G z ) )  /\  ( f `
 0 )  =  ( H `  X
) ) ) `  y )  =  ( ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( X G z ) )  /\  ( f `
 0 )  =  ( H `  X
) ) ) `  Y ) )
10 cvmlift2.k . 2  |-  K  =  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( ( iota_ f  e.  ( II  Cn  C
) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 ) 
|->  ( x G z ) )  /\  (
f `  0 )  =  ( H `  x ) ) ) `
 y ) )
11 fvex 5539 . 2  |-  ( (
iota_ f  e.  (
II  Cn  C )
( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( X G z ) )  /\  ( f `
 0 )  =  ( H `  X
) ) ) `  Y )  e.  _V
128, 9, 10, 11ovmpt2 5983 1  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  e.  ( 0 [,] 1 ) )  ->  ( X K Y )  =  ( ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( X G z ) )  /\  ( f `
 0 )  =  ( H `  X
) ) ) `  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   U.cuni 3827    e. cmpt 4077    o. ccom 4693   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   iota_crio 6297   0cc0 8737   1c1 8738   [,]cicc 10659    Cn ccn 16954    tX ctx 17255   IIcii 18379   CovMap ccvm 23786
This theorem is referenced by:  cvmlift2lem6  23839  cvmlift2lem8  23841
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-reu 2550  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-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-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304
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