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Theorem xpcomco 6331
Description: Composition with the bijection of xpcomf1o 6330 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 6330 . . . . . . . . 9  |-  F :
( A  X.  B
)
-1-1-onto-> ( B  X.  A
)
3 f1ofun 5156 . . . . . . . . 9  |-  ( F : ( A  X.  B ) -1-1-onto-> ( B  X.  A
)  ->  Fun  F )
4 funbrfv2b 5246 . . . . . . . . 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 257 . . . . . . . 8  |-  ( ( u  e.  dom  F  /\  ( F `  u
)  =  w )  <-> 
( ( F `  u )  =  w  /\  u  e.  dom  F ) )
7 eqcom 2058 . . . . . . . . 9  |-  ( ( F `  u )  =  w  <->  w  =  ( F `  u ) )
8 f1odm 5158 . . . . . . . . . . 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 2120 . . . . . . . . 9  |-  ( u  e.  dom  F  <->  u  e.  ( A  X.  B
) )
117, 10anbi12i 441 . . . . . . . 8  |-  ( ( ( F `  u
)  =  w  /\  u  e.  dom  F )  <-> 
( w  =  ( F `  u )  /\  u  e.  ( A  X.  B ) ) )
125, 6, 113bitri 199 . . . . . . 7  |-  ( u F w  <->  ( w  =  ( F `  u )  /\  u  e.  ( A  X.  B
) ) )
1312anbi1i 439 . . . . . 6  |-  ( ( u F w  /\  w G v )  <->  ( (
w  =  ( F `
 u )  /\  u  e.  ( A  X.  B ) )  /\  w G v ) )
14 anass 387 . . . . . 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 177 . . . . 5  |-  ( ( u F w  /\  w G v )  <->  ( w  =  ( F `  u )  /\  (
u  e.  ( A  X.  B )  /\  w G v ) ) )
1615exbii 1512 . . . 4  |-  ( E. w ( u F w  /\  w G v )  <->  E. w
( w  =  ( F `  u )  /\  ( u  e.  ( A  X.  B
)  /\  w G
v ) ) )
17 vex 2577 . . . . . . 7  |-  u  e. 
_V
181mptfvex 5284 . . . . . . 7  |-  ( ( A. x U. `' { x }  e.  _V  /\  u  e.  _V )  ->  ( F `  u )  e.  _V )
1917, 18mpan2 409 . . . . . 6  |-  ( A. x U. `' { x }  e.  _V  ->  ( F `  u )  e.  _V )
20 vex 2577 . . . . . . . . 9  |-  x  e. 
_V
2120snex 3965 . . . . . . . 8  |-  { x }  e.  _V
2221cnvex 4884 . . . . . . 7  |-  `' {
x }  e.  _V
2322uniex 4202 . . . . . 6  |-  U. `' { x }  e.  _V
2419, 23mpg 1356 . . . . 5  |-  ( F `
 u )  e. 
_V
25 breq1 3795 . . . . . 6  |-  ( w  =  ( F `  u )  ->  (
w G v  <->  ( F `  u ) G v ) )
2625anbi2d 445 . . . . 5  |-  ( w  =  ( F `  u )  ->  (
( u  e.  ( A  X.  B )  /\  w G v )  <->  ( u  e.  ( A  X.  B
)  /\  ( F `  u ) G v ) ) )
2724, 26ceqsexv 2610 . . . 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 4390 . . . . . 6  |-  ( u  e.  ( A  X.  B )  <->  E. z E. y ( u  = 
<. z ,  y >.  /\  ( z  e.  A  /\  y  e.  B
) ) )
2928anbi1i 439 . . . . 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 2194 . . . . . . 7  |-  F/_ z
( F `  u
)
31 xpcomco.1 . . . . . . . 8  |-  G  =  ( y  e.  B ,  z  e.  A  |->  C )
32 nfmpt22 5600 . . . . . . . 8  |-  F/_ z
( y  e.  B ,  z  e.  A  |->  C )
3331, 32nfcxfr 2191 . . . . . . 7  |-  F/_ z G
34 nfcv 2194 . . . . . . 7  |-  F/_ z
v
3530, 33, 34nfbr 3836 . . . . . 6  |-  F/ z ( F `  u
) G v
363519.41 1592 . . . . 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 2194 . . . . . . . . 9  |-  F/_ y
( F `  u
)
38 nfmpt21 5599 . . . . . . . . . 10  |-  F/_ y
( y  e.  B ,  z  e.  A  |->  C )
3931, 38nfcxfr 2191 . . . . . . . . 9  |-  F/_ y G
40 nfcv 2194 . . . . . . . . 9  |-  F/_ y
v
4137, 39, 40nfbr 3836 . . . . . . . 8  |-  F/ y ( F `  u
) G v
424119.41 1592 . . . . . . 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 387 . . . . . . . . 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 5206 . . . . . . . . . . . . . 14  |-  ( u  =  <. z ,  y
>.  ->  ( F `  u )  =  ( F `  <. z ,  y >. )
)
45 opelxpi 4404 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  A  /\  y  e.  B )  -> 
<. z ,  y >.  e.  ( A  X.  B
) )
46 sneq 3414 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  <. z ,  y
>.  ->  { x }  =  { <. z ,  y
>. } )
4746cnveqd 4539 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  <. z ,  y
>.  ->  `' { x }  =  `' { <. z ,  y >. } )
4847unieqd 3619 . . . . . . . . . . . . . . . . 17  |-  ( x  =  <. z ,  y
>.  ->  U. `' { x }  =  U. `' { <. z ,  y >. } )
49 vex 2577 . . . . . . . . . . . . . . . . . 18  |-  z  e. 
_V
50 vex 2577 . . . . . . . . . . . . . . . . . 18  |-  y  e. 
_V
51 opswapg 4835 . . . . . . . . . . . . . . . . . 18  |-  ( ( z  e.  _V  /\  y  e.  _V )  ->  U. `' { <. z ,  y >. }  =  <. y ,  z >.
)
5249, 50, 51mp2an 410 . . . . . . . . . . . . . . . . 17  |-  U. `' { <. z ,  y
>. }  =  <. y ,  z >.
5348, 52syl6eq 2104 . . . . . . . . . . . . . . . 16  |-  ( x  =  <. z ,  y
>.  ->  U. `' { x }  =  <. y ,  z >. )
5450, 49opex 3994 . . . . . . . . . . . . . . . 16  |-  <. y ,  z >.  e.  _V
5553, 1, 54fvmpt 5277 . . . . . . . . . . . . . . 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 2108 . . . . . . . . . . . . 13  |-  ( ( u  =  <. z ,  y >.  /\  (
z  e.  A  /\  y  e.  B )
)  ->  ( F `  u )  =  <. y ,  z >. )
5857breq1d 3802 . . . . . . . . . . . 12  |-  ( ( u  =  <. z ,  y >.  /\  (
z  e.  A  /\  y  e.  B )
)  ->  ( ( F `  u ) G v  <->  <. y ,  z >. G v ) )
59 df-br 3793 . . . . . . . . . . . . . . . 16  |-  ( <.
y ,  z >. G v  <->  <. <. y ,  z >. ,  v
>.  e.  G )
60 df-mpt2 5545 . . . . . . . . . . . . . . . . . 18  |-  ( y  e.  B ,  z  e.  A  |->  C )  =  { <. <. y ,  z >. ,  v
>.  |  ( (
y  e.  B  /\  z  e.  A )  /\  v  =  C
) }
6131, 60eqtri 2076 . . . . . . . . . . . . . . . . 17  |-  G  =  { <. <. y ,  z
>. ,  v >.  |  ( ( y  e.  B  /\  z  e.  A )  /\  v  =  C ) }
6261eleq2i 2120 . . . . . . . . . . . . . . . 16  |-  ( <. <. y ,  z >. ,  v >.  e.  G  <->  <. <. y ,  z >. ,  v >.  e.  { <. <. y ,  z
>. ,  v >.  |  ( ( y  e.  B  /\  z  e.  A )  /\  v  =  C ) } )
63 oprabid 5565 . . . . . . . . . . . . . . . 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 199 . . . . . . . . . . . . . . 15  |-  ( <.
y ,  z >. G v  <->  ( (
y  e.  B  /\  z  e.  A )  /\  v  =  C
) )
6564baib 839 . . . . . . . . . . . . . 14  |-  ( ( y  e.  B  /\  z  e.  A )  ->  ( <. y ,  z
>. G v  <->  v  =  C ) )
6665ancoms 259 . . . . . . . . . . . . 13  |-  ( ( z  e.  A  /\  y  e.  B )  ->  ( <. y ,  z
>. G v  <->  v  =  C ) )
6766adantl 266 . . . . . . . . . . . 12  |-  ( ( u  =  <. z ,  y >.  /\  (
z  e.  A  /\  y  e.  B )
)  ->  ( <. y ,  z >. G v  <-> 
v  =  C ) )
6858, 67bitrd 181 . . . . . . . . . . 11  |-  ( ( u  =  <. z ,  y >.  /\  (
z  e.  A  /\  y  e.  B )
)  ->  ( ( F `  u ) G v  <->  v  =  C ) )
6968pm5.32da 433 . . . . . . . . . 10  |-  ( u  =  <. z ,  y
>.  ->  ( ( ( z  e.  A  /\  y  e.  B )  /\  ( F `  u
) G v )  <-> 
( ( z  e.  A  /\  y  e.  B )  /\  v  =  C ) ) )
7069pm5.32i 435 . . . . . . . . 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 177 . . . . . . . 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 1512 . . . . . . 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 179 . . . . . 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 1512 . . . . 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 201 . . . 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 199 . . 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 3852 . 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 4382 . 2  |-  ( G  o.  F )  =  { <. u ,  v
>.  |  E. w
( u F w  /\  w G v ) }
79 df-mpt2 5545 . . 3  |-  ( z  e.  A ,  y  e.  B  |->  C )  =  { <. <. z ,  y >. ,  v
>.  |  ( (
z  e.  A  /\  y  e.  B )  /\  v  =  C
) }
80 dfoprab2 5580 . . 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 2076 . 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 2086 1  |-  ( G  o.  F )  =  ( z  e.  A ,  y  e.  B  |->  C )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102   A.wal 1257    = wceq 1259   E.wex 1397    e. wcel 1409   _Vcvv 2574   {csn 3403   <.cop 3406   U.cuni 3608   class class class wbr 3792   {copab 3845    |-> cmpt 3846    X. cxp 4371   `'ccnv 4372   dom cdm 4373    o. ccom 4377   Fun wfun 4924   -1-1-onto->wf1o 4929   ` cfv 4930   {coprab 5541    |-> cmpt2 5542
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796
This theorem is referenced by: (None)
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