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Theorem xpcomco 6487
Description: Composition with the bijection of xpcomf1o 6486 swaps the arguments to a mapping. (Contributed by Mario Carneiro, 30-May-2015.)
Hypotheses
Ref Expression
xpcomf1o.1  |-  F  =  ( x  e.  ( A  X.  B ) 
|->  U. `' { x } )
xpcomco.1  |-  G  =  ( y  e.  B ,  z  e.  A  |->  C )
Assertion
Ref Expression
xpcomco  |-  ( G  o.  F )  =  ( z  e.  A ,  y  e.  B  |->  C )
Distinct variable groups:    x, y, z, A    x, B, y, z    y, F, z
Allowed substitution hints:    C( x, y, z)    F( x)    G( x, y, z)

Proof of Theorem xpcomco
Dummy variables  v  u  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpcomf1o.1 . . . . . . . . . 10  |-  F  =  ( x  e.  ( A  X.  B ) 
|->  U. `' { x } )
21xpcomf1o 6486 . . . . . . . . 9  |-  F :
( A  X.  B
)
-1-1-onto-> ( B  X.  A
)
3 f1ofun 5211 . . . . . . . . 9  |-  ( F : ( A  X.  B ) -1-1-onto-> ( B  X.  A
)  ->  Fun  F )
4 funbrfv2b 5305 . . . . . . . . 9  |-  ( Fun 
F  ->  ( u F w  <->  ( u  e. 
dom  F  /\  ( F `  u )  =  w ) ) )
52, 3, 4mp2b 8 . . . . . . . 8  |-  ( u F w  <->  ( u  e.  dom  F  /\  ( F `  u )  =  w ) )
6 ancom 262 . . . . . . . 8  |-  ( ( u  e.  dom  F  /\  ( F `  u
)  =  w )  <-> 
( ( F `  u )  =  w  /\  u  e.  dom  F ) )
7 eqcom 2087 . . . . . . . . 9  |-  ( ( F `  u )  =  w  <->  w  =  ( F `  u ) )
8 f1odm 5213 . . . . . . . . . . 11  |-  ( F : ( A  X.  B ) -1-1-onto-> ( B  X.  A
)  ->  dom  F  =  ( A  X.  B
) )
92, 8ax-mp 7 . . . . . . . . . 10  |-  dom  F  =  ( A  X.  B )
109eleq2i 2151 . . . . . . . . 9  |-  ( u  e.  dom  F  <->  u  e.  ( A  X.  B
) )
117, 10anbi12i 448 . . . . . . . 8  |-  ( ( ( F `  u
)  =  w  /\  u  e.  dom  F )  <-> 
( w  =  ( F `  u )  /\  u  e.  ( A  X.  B ) ) )
125, 6, 113bitri 204 . . . . . . 7  |-  ( u F w  <->  ( w  =  ( F `  u )  /\  u  e.  ( A  X.  B
) ) )
1312anbi1i 446 . . . . . 6  |-  ( ( u F w  /\  w G v )  <->  ( (
w  =  ( F `
 u )  /\  u  e.  ( A  X.  B ) )  /\  w G v ) )
14 anass 393 . . . . . 6  |-  ( ( ( w  =  ( F `  u )  /\  u  e.  ( A  X.  B ) )  /\  w G v )  <->  ( w  =  ( F `  u )  /\  (
u  e.  ( A  X.  B )  /\  w G v ) ) )
1513, 14bitri 182 . . . . 5  |-  ( ( u F w  /\  w G v )  <->  ( w  =  ( F `  u )  /\  (
u  e.  ( A  X.  B )  /\  w G v ) ) )
1615exbii 1539 . . . 4  |-  ( E. w ( u F w  /\  w G v )  <->  E. w
( w  =  ( F `  u )  /\  ( u  e.  ( A  X.  B
)  /\  w G
v ) ) )
17 vex 2618 . . . . . . 7  |-  u  e. 
_V
181mptfvex 5344 . . . . . . 7  |-  ( ( A. x U. `' { x }  e.  _V  /\  u  e.  _V )  ->  ( F `  u )  e.  _V )
1917, 18mpan2 416 . . . . . 6  |-  ( A. x U. `' { x }  e.  _V  ->  ( F `  u )  e.  _V )
20 vex 2618 . . . . . . . . 9  |-  x  e. 
_V
2120snex 3992 . . . . . . . 8  |-  { x }  e.  _V
2221cnvex 4931 . . . . . . 7  |-  `' {
x }  e.  _V
2322uniex 4236 . . . . . 6  |-  U. `' { x }  e.  _V
2419, 23mpg 1383 . . . . 5  |-  ( F `
 u )  e. 
_V
25 breq1 3822 . . . . . 6  |-  ( w  =  ( F `  u )  ->  (
w G v  <->  ( F `  u ) G v ) )
2625anbi2d 452 . . . . 5  |-  ( w  =  ( F `  u )  ->  (
( u  e.  ( A  X.  B )  /\  w G v )  <->  ( u  e.  ( A  X.  B
)  /\  ( F `  u ) G v ) ) )
2724, 26ceqsexv 2652 . . . 4  |-  ( E. w ( w  =  ( F `  u
)  /\  ( u  e.  ( A  X.  B
)  /\  w G
v ) )  <->  ( u  e.  ( A  X.  B
)  /\  ( F `  u ) G v ) )
28 elxp 4426 . . . . . 6  |-  ( u  e.  ( A  X.  B )  <->  E. z E. y ( u  = 
<. z ,  y >.  /\  ( z  e.  A  /\  y  e.  B
) ) )
2928anbi1i 446 . . . . 5  |-  ( ( u  e.  ( A  X.  B )  /\  ( F `  u ) G v )  <->  ( E. z E. y ( u  =  <. z ,  y
>.  /\  ( z  e.  A  /\  y  e.  B ) )  /\  ( F `  u ) G v ) )
30 nfcv 2225 . . . . . . 7  |-  F/_ z
( F `  u
)
31 xpcomco.1 . . . . . . . 8  |-  G  =  ( y  e.  B ,  z  e.  A  |->  C )
32 nfmpt22 5666 . . . . . . . 8  |-  F/_ z
( y  e.  B ,  z  e.  A  |->  C )
3331, 32nfcxfr 2222 . . . . . . 7  |-  F/_ z G
34 nfcv 2225 . . . . . . 7  |-  F/_ z
v
3530, 33, 34nfbr 3863 . . . . . 6  |-  F/ z ( F `  u
) G v
363519.41 1619 . . . . 5  |-  ( E. z ( E. y
( u  =  <. z ,  y >.  /\  (
z  e.  A  /\  y  e.  B )
)  /\  ( F `  u ) G v )  <->  ( E. z E. y ( u  = 
<. z ,  y >.  /\  ( z  e.  A  /\  y  e.  B
) )  /\  ( F `  u ) G v ) )
37 nfcv 2225 . . . . . . . . 9  |-  F/_ y
( F `  u
)
38 nfmpt21 5665 . . . . . . . . . 10  |-  F/_ y
( y  e.  B ,  z  e.  A  |->  C )
3931, 38nfcxfr 2222 . . . . . . . . 9  |-  F/_ y G
40 nfcv 2225 . . . . . . . . 9  |-  F/_ y
v
4137, 39, 40nfbr 3863 . . . . . . . 8  |-  F/ y ( F `  u
) G v
424119.41 1619 . . . . . . 7  |-  ( E. y ( ( u  =  <. z ,  y
>.  /\  ( z  e.  A  /\  y  e.  B ) )  /\  ( F `  u ) G v )  <->  ( E. y ( u  = 
<. z ,  y >.  /\  ( z  e.  A  /\  y  e.  B
) )  /\  ( F `  u ) G v ) )
43 anass 393 . . . . . . . . 9  |-  ( ( ( u  =  <. z ,  y >.  /\  (
z  e.  A  /\  y  e.  B )
)  /\  ( F `  u ) G v )  <->  ( u  = 
<. z ,  y >.  /\  ( ( z  e.  A  /\  y  e.  B )  /\  ( F `  u ) G v ) ) )
44 fveq2 5261 . . . . . . . . . . . . . 14  |-  ( u  =  <. z ,  y
>.  ->  ( F `  u )  =  ( F `  <. z ,  y >. )
)
45 opelxpi 4440 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  A  /\  y  e.  B )  -> 
<. z ,  y >.  e.  ( A  X.  B
) )
46 sneq 3441 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  <. z ,  y
>.  ->  { x }  =  { <. z ,  y
>. } )
4746cnveqd 4578 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  <. z ,  y
>.  ->  `' { x }  =  `' { <. z ,  y >. } )
4847unieqd 3646 . . . . . . . . . . . . . . . . 17  |-  ( x  =  <. z ,  y
>.  ->  U. `' { x }  =  U. `' { <. z ,  y >. } )
49 vex 2618 . . . . . . . . . . . . . . . . . 18  |-  z  e. 
_V
50 vex 2618 . . . . . . . . . . . . . . . . . 18  |-  y  e. 
_V
51 opswapg 4879 . . . . . . . . . . . . . . . . . 18  |-  ( ( z  e.  _V  /\  y  e.  _V )  ->  U. `' { <. z ,  y >. }  =  <. y ,  z >.
)
5249, 50, 51mp2an 417 . . . . . . . . . . . . . . . . 17  |-  U. `' { <. z ,  y
>. }  =  <. y ,  z >.
5348, 52syl6eq 2133 . . . . . . . . . . . . . . . 16  |-  ( x  =  <. z ,  y
>.  ->  U. `' { x }  =  <. y ,  z >. )
5450, 49opex 4028 . . . . . . . . . . . . . . . 16  |-  <. y ,  z >.  e.  _V
5553, 1, 54fvmpt 5337 . . . . . . . . . . . . . . 15  |-  ( <.
z ,  y >.  e.  ( A  X.  B
)  ->  ( F `  <. z ,  y
>. )  =  <. y ,  z >. )
5645, 55syl 14 . . . . . . . . . . . . . 14  |-  ( ( z  e.  A  /\  y  e.  B )  ->  ( F `  <. z ,  y >. )  =  <. y ,  z
>. )
5744, 56sylan9eq 2137 . . . . . . . . . . . . 13  |-  ( ( u  =  <. z ,  y >.  /\  (
z  e.  A  /\  y  e.  B )
)  ->  ( F `  u )  =  <. y ,  z >. )
5857breq1d 3829 . . . . . . . . . . . 12  |-  ( ( u  =  <. z ,  y >.  /\  (
z  e.  A  /\  y  e.  B )
)  ->  ( ( F `  u ) G v  <->  <. y ,  z >. G v ) )
59 df-br 3820 . . . . . . . . . . . . . . . 16  |-  ( <.
y ,  z >. G v  <->  <. <. y ,  z >. ,  v
>.  e.  G )
60 df-mpt2 5611 . . . . . . . . . . . . . . . . . 18  |-  ( y  e.  B ,  z  e.  A  |->  C )  =  { <. <. y ,  z >. ,  v
>.  |  ( (
y  e.  B  /\  z  e.  A )  /\  v  =  C
) }
6131, 60eqtri 2105 . . . . . . . . . . . . . . . . 17  |-  G  =  { <. <. y ,  z
>. ,  v >.  |  ( ( y  e.  B  /\  z  e.  A )  /\  v  =  C ) }
6261eleq2i 2151 . . . . . . . . . . . . . . . 16  |-  ( <. <. y ,  z >. ,  v >.  e.  G  <->  <. <. y ,  z >. ,  v >.  e.  { <. <. y ,  z
>. ,  v >.  |  ( ( y  e.  B  /\  z  e.  A )  /\  v  =  C ) } )
63 oprabid 5631 . . . . . . . . . . . . . . . 16  |-  ( <. <. y ,  z >. ,  v >.  e.  { <. <. y ,  z
>. ,  v >.  |  ( ( y  e.  B  /\  z  e.  A )  /\  v  =  C ) }  <->  ( (
y  e.  B  /\  z  e.  A )  /\  v  =  C
) )
6459, 62, 633bitri 204 . . . . . . . . . . . . . . 15  |-  ( <.
y ,  z >. G v  <->  ( (
y  e.  B  /\  z  e.  A )  /\  v  =  C
) )
6564baib 864 . . . . . . . . . . . . . 14  |-  ( ( y  e.  B  /\  z  e.  A )  ->  ( <. y ,  z
>. G v  <->  v  =  C ) )
6665ancoms 264 . . . . . . . . . . . . 13  |-  ( ( z  e.  A  /\  y  e.  B )  ->  ( <. y ,  z
>. G v  <->  v  =  C ) )
6766adantl 271 . . . . . . . . . . . 12  |-  ( ( u  =  <. z ,  y >.  /\  (
z  e.  A  /\  y  e.  B )
)  ->  ( <. y ,  z >. G v  <-> 
v  =  C ) )
6858, 67bitrd 186 . . . . . . . . . . 11  |-  ( ( u  =  <. z ,  y >.  /\  (
z  e.  A  /\  y  e.  B )
)  ->  ( ( F `  u ) G v  <->  v  =  C ) )
6968pm5.32da 440 . . . . . . . . . 10  |-  ( u  =  <. z ,  y
>.  ->  ( ( ( z  e.  A  /\  y  e.  B )  /\  ( F `  u
) G v )  <-> 
( ( z  e.  A  /\  y  e.  B )  /\  v  =  C ) ) )
7069pm5.32i 442 . . . . . . . . 9  |-  ( ( u  =  <. z ,  y >.  /\  (
( z  e.  A  /\  y  e.  B
)  /\  ( F `  u ) G v ) )  <->  ( u  =  <. z ,  y
>.  /\  ( ( z  e.  A  /\  y  e.  B )  /\  v  =  C ) ) )
7143, 70bitri 182 . . . . . . . 8  |-  ( ( ( u  =  <. z ,  y >.  /\  (
z  e.  A  /\  y  e.  B )
)  /\  ( F `  u ) G v )  <->  ( u  = 
<. z ,  y >.  /\  ( ( z  e.  A  /\  y  e.  B )  /\  v  =  C ) ) )
7271exbii 1539 . . . . . . 7  |-  ( E. y ( ( u  =  <. z ,  y
>.  /\  ( z  e.  A  /\  y  e.  B ) )  /\  ( F `  u ) G v )  <->  E. y
( u  =  <. z ,  y >.  /\  (
( z  e.  A  /\  y  e.  B
)  /\  v  =  C ) ) )
7342, 72bitr3i 184 . . . . . 6  |-  ( ( E. y ( u  =  <. z ,  y
>.  /\  ( z  e.  A  /\  y  e.  B ) )  /\  ( F `  u ) G v )  <->  E. y
( u  =  <. z ,  y >.  /\  (
( z  e.  A  /\  y  e.  B
)  /\  v  =  C ) ) )
7473exbii 1539 . . . . 5  |-  ( E. z ( E. y
( u  =  <. z ,  y >.  /\  (
z  e.  A  /\  y  e.  B )
)  /\  ( F `  u ) G v )  <->  E. z E. y
( u  =  <. z ,  y >.  /\  (
( z  e.  A  /\  y  e.  B
)  /\  v  =  C ) ) )
7529, 36, 743bitr2i 206 . . . 4  |-  ( ( u  e.  ( A  X.  B )  /\  ( F `  u ) G v )  <->  E. z E. y ( u  = 
<. z ,  y >.  /\  ( ( z  e.  A  /\  y  e.  B )  /\  v  =  C ) ) )
7616, 27, 753bitri 204 . . 3  |-  ( E. w ( u F w  /\  w G v )  <->  E. z E. y ( u  = 
<. z ,  y >.  /\  ( ( z  e.  A  /\  y  e.  B )  /\  v  =  C ) ) )
7776opabbii 3879 . 2  |-  { <. u ,  v >.  |  E. w ( u F w  /\  w G v ) }  =  { <. u ,  v
>.  |  E. z E. y ( u  = 
<. z ,  y >.  /\  ( ( z  e.  A  /\  y  e.  B )  /\  v  =  C ) ) }
78 df-co 4418 . 2  |-  ( G  o.  F )  =  { <. u ,  v
>.  |  E. w
( u F w  /\  w G v ) }
79 df-mpt2 5611 . . 3  |-  ( z  e.  A ,  y  e.  B  |->  C )  =  { <. <. z ,  y >. ,  v
>.  |  ( (
z  e.  A  /\  y  e.  B )  /\  v  =  C
) }
80 dfoprab2 5646 . . 3  |-  { <. <.
z ,  y >. ,  v >.  |  ( ( z  e.  A  /\  y  e.  B
)  /\  v  =  C ) }  =  { <. u ,  v
>.  |  E. z E. y ( u  = 
<. z ,  y >.  /\  ( ( z  e.  A  /\  y  e.  B )  /\  v  =  C ) ) }
8179, 80eqtri 2105 . 2  |-  ( z  e.  A ,  y  e.  B  |->  C )  =  { <. u ,  v >.  |  E. z E. y ( u  =  <. z ,  y
>.  /\  ( ( z  e.  A  /\  y  e.  B )  /\  v  =  C ) ) }
8277, 78, 813eqtr4i 2115 1  |-  ( G  o.  F )  =  ( z  e.  A ,  y  e.  B  |->  C )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103   A.wal 1285    = wceq 1287   E.wex 1424    e. wcel 1436   _Vcvv 2615   {csn 3430   <.cop 3433   U.cuni 3635   class class class wbr 3819   {copab 3872    |-> cmpt 3873    X. cxp 4407   `'ccnv 4408   dom cdm 4409    o. ccom 4413   Fun wfun 4971   -1-1-onto->wf1o 4976   ` cfv 4977   {coprab 5607    |-> cmpt2 5608
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3930  ax-pow 3982  ax-pr 4008  ax-un 4232  ax-setind 4324
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-pw 3416  df-sn 3436  df-pr 3437  df-op 3439  df-uni 3636  df-br 3820  df-opab 3874  df-mpt 3875  df-id 4092  df-xp 4415  df-rel 4416  df-cnv 4417  df-co 4418  df-dm 4419  df-rn 4420  df-iota 4942  df-fun 4979  df-fn 4980  df-f 4981  df-f1 4982  df-fo 4983  df-f1o 4984  df-fv 4985  df-oprab 5610  df-mpt2 5611  df-1st 5861  df-2nd 5862
This theorem is referenced by: (None)
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