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Theorem xpcomco 7190
Description: Composition with the bijection of xpcomf1o 7189 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 7189 . . . . . . . . 9  |-  F :
( A  X.  B
)
-1-1-onto-> ( B  X.  A
)
3 f1ofun 5668 . . . . . . . . 9  |-  ( F : ( A  X.  B ) -1-1-onto-> ( B  X.  A
)  ->  Fun  F )
4 funbrfv2b 5763 . . . . . . . . 9  |-  ( Fun 
F  ->  ( u F w  <->  ( u  e. 
dom  F  /\  ( F `  u )  =  w ) ) )
52, 3, 4mp2b 10 . . . . . . . 8  |-  ( u F w  <->  ( u  e.  dom  F  /\  ( F `  u )  =  w ) )
6 ancom 438 . . . . . . . 8  |-  ( ( u  e.  dom  F  /\  ( F `  u
)  =  w )  <-> 
( ( F `  u )  =  w  /\  u  e.  dom  F ) )
7 eqcom 2437 . . . . . . . . 9  |-  ( ( F `  u )  =  w  <->  w  =  ( F `  u ) )
8 f1odm 5670 . . . . . . . . . . 11  |-  ( F : ( A  X.  B ) -1-1-onto-> ( B  X.  A
)  ->  dom  F  =  ( A  X.  B
) )
92, 8ax-mp 8 . . . . . . . . . 10  |-  dom  F  =  ( A  X.  B )
109eleq2i 2499 . . . . . . . . 9  |-  ( u  e.  dom  F  <->  u  e.  ( A  X.  B
) )
117, 10anbi12i 679 . . . . . . . 8  |-  ( ( ( F `  u
)  =  w  /\  u  e.  dom  F )  <-> 
( w  =  ( F `  u )  /\  u  e.  ( A  X.  B ) ) )
125, 6, 113bitri 263 . . . . . . 7  |-  ( u F w  <->  ( w  =  ( F `  u )  /\  u  e.  ( A  X.  B
) ) )
1312anbi1i 677 . . . . . 6  |-  ( ( u F w  /\  w G v )  <->  ( (
w  =  ( F `
 u )  /\  u  e.  ( A  X.  B ) )  /\  w G v ) )
14 anass 631 . . . . . 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 241 . . . . 5  |-  ( ( u F w  /\  w G v )  <->  ( w  =  ( F `  u )  /\  (
u  e.  ( A  X.  B )  /\  w G v ) ) )
1615exbii 1592 . . . 4  |-  ( E. w ( u F w  /\  w G v )  <->  E. w
( w  =  ( F `  u )  /\  ( u  e.  ( A  X.  B
)  /\  w G
v ) ) )
17 fvex 5734 . . . . 5  |-  ( F `
 u )  e. 
_V
18 breq1 4207 . . . . . 6  |-  ( w  =  ( F `  u )  ->  (
w G v  <->  ( F `  u ) G v ) )
1918anbi2d 685 . . . . 5  |-  ( w  =  ( F `  u )  ->  (
( u  e.  ( A  X.  B )  /\  w G v )  <->  ( u  e.  ( A  X.  B
)  /\  ( F `  u ) G v ) ) )
2017, 19ceqsexv 2983 . . . 4  |-  ( E. w ( w  =  ( F `  u
)  /\  ( u  e.  ( A  X.  B
)  /\  w G
v ) )  <->  ( u  e.  ( A  X.  B
)  /\  ( F `  u ) G v ) )
21 elxp 4887 . . . . . 6  |-  ( u  e.  ( A  X.  B )  <->  E. z E. y ( u  = 
<. z ,  y >.  /\  ( z  e.  A  /\  y  e.  B
) ) )
2221anbi1i 677 . . . . 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 ) )
23 nfcv 2571 . . . . . . 7  |-  F/_ z
( F `  u
)
24 xpcomco.1 . . . . . . . 8  |-  G  =  ( y  e.  B ,  z  e.  A  |->  C )
25 nfmpt22 6133 . . . . . . . 8  |-  F/_ z
( y  e.  B ,  z  e.  A  |->  C )
2624, 25nfcxfr 2568 . . . . . . 7  |-  F/_ z G
27 nfcv 2571 . . . . . . 7  |-  F/_ z
v
2823, 26, 27nfbr 4248 . . . . . 6  |-  F/ z ( F `  u
) G v
292819.41 1900 . . . . 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 ) )
30 nfcv 2571 . . . . . . . . 9  |-  F/_ y
( F `  u
)
31 nfmpt21 6132 . . . . . . . . . 10  |-  F/_ y
( y  e.  B ,  z  e.  A  |->  C )
3224, 31nfcxfr 2568 . . . . . . . . 9  |-  F/_ y G
33 nfcv 2571 . . . . . . . . 9  |-  F/_ y
v
3430, 32, 33nfbr 4248 . . . . . . . 8  |-  F/ y ( F `  u
) G v
353419.41 1900 . . . . . . 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 ) )
36 anass 631 . . . . . . . . 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 ) ) )
37 fveq2 5720 . . . . . . . . . . . . . 14  |-  ( u  =  <. z ,  y
>.  ->  ( F `  u )  =  ( F `  <. z ,  y >. )
)
38 opelxpi 4902 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  A  /\  y  e.  B )  -> 
<. z ,  y >.  e.  ( A  X.  B
) )
39 sneq 3817 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  <. z ,  y
>.  ->  { x }  =  { <. z ,  y
>. } )
4039cnveqd 5040 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  <. z ,  y
>.  ->  `' { x }  =  `' { <. z ,  y >. } )
4140unieqd 4018 . . . . . . . . . . . . . . . . 17  |-  ( x  =  <. z ,  y
>.  ->  U. `' { x }  =  U. `' { <. z ,  y >. } )
42 opswap 5348 . . . . . . . . . . . . . . . . 17  |-  U. `' { <. z ,  y
>. }  =  <. y ,  z >.
4341, 42syl6eq 2483 . . . . . . . . . . . . . . . 16  |-  ( x  =  <. z ,  y
>.  ->  U. `' { x }  =  <. y ,  z >. )
44 opex 4419 . . . . . . . . . . . . . . . 16  |-  <. y ,  z >.  e.  _V
4543, 1, 44fvmpt 5798 . . . . . . . . . . . . . . 15  |-  ( <.
z ,  y >.  e.  ( A  X.  B
)  ->  ( F `  <. z ,  y
>. )  =  <. y ,  z >. )
4638, 45syl 16 . . . . . . . . . . . . . 14  |-  ( ( z  e.  A  /\  y  e.  B )  ->  ( F `  <. z ,  y >. )  =  <. y ,  z
>. )
4737, 46sylan9eq 2487 . . . . . . . . . . . . 13  |-  ( ( u  =  <. z ,  y >.  /\  (
z  e.  A  /\  y  e.  B )
)  ->  ( F `  u )  =  <. y ,  z >. )
4847breq1d 4214 . . . . . . . . . . . 12  |-  ( ( u  =  <. z ,  y >.  /\  (
z  e.  A  /\  y  e.  B )
)  ->  ( ( F `  u ) G v  <->  <. y ,  z >. G v ) )
49 df-br 4205 . . . . . . . . . . . . . . . 16  |-  ( <.
y ,  z >. G v  <->  <. <. y ,  z >. ,  v
>.  e.  G )
50 df-mpt2 6078 . . . . . . . . . . . . . . . . . 18  |-  ( y  e.  B ,  z  e.  A  |->  C )  =  { <. <. y ,  z >. ,  v
>.  |  ( (
y  e.  B  /\  z  e.  A )  /\  v  =  C
) }
5124, 50eqtri 2455 . . . . . . . . . . . . . . . . 17  |-  G  =  { <. <. y ,  z
>. ,  v >.  |  ( ( y  e.  B  /\  z  e.  A )  /\  v  =  C ) }
5251eleq2i 2499 . . . . . . . . . . . . . . . 16  |-  ( <. <. y ,  z >. ,  v >.  e.  G  <->  <. <. y ,  z >. ,  v >.  e.  { <. <. y ,  z
>. ,  v >.  |  ( ( y  e.  B  /\  z  e.  A )  /\  v  =  C ) } )
53 oprabid 6097 . . . . . . . . . . . . . . . 16  |-  ( <. <. y ,  z >. ,  v >.  e.  { <. <. y ,  z
>. ,  v >.  |  ( ( y  e.  B  /\  z  e.  A )  /\  v  =  C ) }  <->  ( (
y  e.  B  /\  z  e.  A )  /\  v  =  C
) )
5449, 52, 533bitri 263 . . . . . . . . . . . . . . 15  |-  ( <.
y ,  z >. G v  <->  ( (
y  e.  B  /\  z  e.  A )  /\  v  =  C
) )
5554baib 872 . . . . . . . . . . . . . 14  |-  ( ( y  e.  B  /\  z  e.  A )  ->  ( <. y ,  z
>. G v  <->  v  =  C ) )
5655ancoms 440 . . . . . . . . . . . . 13  |-  ( ( z  e.  A  /\  y  e.  B )  ->  ( <. y ,  z
>. G v  <->  v  =  C ) )
5756adantl 453 . . . . . . . . . . . 12  |-  ( ( u  =  <. z ,  y >.  /\  (
z  e.  A  /\  y  e.  B )
)  ->  ( <. y ,  z >. G v  <-> 
v  =  C ) )
5848, 57bitrd 245 . . . . . . . . . . 11  |-  ( ( u  =  <. z ,  y >.  /\  (
z  e.  A  /\  y  e.  B )
)  ->  ( ( F `  u ) G v  <->  v  =  C ) )
5958pm5.32da 623 . . . . . . . . . 10  |-  ( u  =  <. z ,  y
>.  ->  ( ( ( z  e.  A  /\  y  e.  B )  /\  ( F `  u
) G v )  <-> 
( ( z  e.  A  /\  y  e.  B )  /\  v  =  C ) ) )
6059pm5.32i 619 . . . . . . . . 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 ) ) )
6136, 60bitri 241 . . . . . . . 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 ) ) )
6261exbii 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
)  /\  v  =  C ) ) )
6335, 62bitr3i 243 . . . . . 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 ) ) )
6463exbii 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
)  /\  v  =  C ) ) )
6522, 29, 643bitr2i 265 . . . 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 ) ) )
6616, 20, 653bitri 263 . . 3  |-  ( E. w ( u F w  /\  w G v )  <->  E. z E. y ( u  = 
<. z ,  y >.  /\  ( ( z  e.  A  /\  y  e.  B )  /\  v  =  C ) ) )
6766opabbii 4264 . 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 ) ) }
68 df-co 4879 . 2  |-  ( G  o.  F )  =  { <. u ,  v
>.  |  E. w
( u F w  /\  w G v ) }
69 df-mpt2 6078 . . 3  |-  ( z  e.  A ,  y  e.  B  |->  C )  =  { <. <. z ,  y >. ,  v
>.  |  ( (
z  e.  A  /\  y  e.  B )  /\  v  =  C
) }
70 dfoprab2 6113 . . 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 ) ) }
7169, 70eqtri 2455 . 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 ) ) }
7267, 68, 713eqtr4i 2465 1  |-  ( G  o.  F )  =  ( z  e.  A ,  y  e.  B  |->  C )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   {csn 3806   <.cop 3809   U.cuni 4007   class class class wbr 4204   {copab 4257    e. cmpt 4258    X. cxp 4868   `'ccnv 4869   dom cdm 4870    o. ccom 4874   Fun wfun 5440   -1-1-onto->wf1o 5445   ` cfv 5446   {coprab 6074    e. cmpt2 6075
This theorem is referenced by:  omf1o  7203
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342
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