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Theorem cvmlift2lem4 24994
Description: Lemma for cvmlift2 25004. (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 6089 . . . . . . 7  |-  ( x  =  X  ->  (
x G z )  =  ( X G z ) )
21mpteq2dv 4297 . . . . . 6  |-  ( x  =  X  ->  (
z  e.  ( 0 [,] 1 )  |->  ( x G z ) )  =  ( z  e.  ( 0 [,] 1 )  |->  ( X G z ) ) )
32eqeq2d 2448 . . . . 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 5729 . . . . . 6  |-  ( x  =  X  ->  ( H `  x )  =  ( H `  X ) )
54eqeq2d 2448 . . . . 5  |-  ( x  =  X  ->  (
( f `  0
)  =  ( H `
 x )  <->  ( f `  0 )  =  ( H `  X
) ) )
63, 5anbi12d 693 . . . 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 6552 . . 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 5731 . 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 5729 . 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 5743 . 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 6210 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 360    = wceq 1653    e. wcel 1726   U.cuni 4016    e. cmpt 4267    o. ccom 4883   ` cfv 5455  (class class class)co 6082    e. cmpt2 6084   iota_crio 6543   0cc0 8991   1c1 8992   [,]cicc 10920    Cn ccn 17289    tX ctx 17593   IIcii 18906   CovMap ccvm 24943
This theorem is referenced by:  cvmlift2lem6  24996  cvmlift2lem8  24998
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 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-iota 5419  df-fun 5457  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-riota 6550
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