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Theorem tposmpt2 6518
Description: Transposition of a two-argument mapping. (Contributed by Mario Carneiro, 10-Sep-2015.)
Hypothesis
Ref Expression
tposmpt2.1  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
Assertion
Ref Expression
tposmpt2  |- tpos  F  =  ( y  e.  B ,  x  e.  A  |->  C )
Distinct variable group:    x, y
Allowed substitution hints:    A( x, y)    B( x, y)    C( x, y)    F( x, y)

Proof of Theorem tposmpt2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 tposmpt2.1 . . . 4  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
2 df-mpt2 6088 . . . 4  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
3 ancom 439 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  B )  <->  ( y  e.  B  /\  x  e.  A )
)
43anbi1i 678 . . . . 5  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  z  =  C )  <->  ( (
y  e.  B  /\  x  e.  A )  /\  z  =  C
) )
54oprabbii 6131 . . . 4  |-  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B
)  /\  z  =  C ) }  =  { <. <. x ,  y
>. ,  z >.  |  ( ( y  e.  B  /\  x  e.  A )  /\  z  =  C ) }
61, 2, 53eqtri 2462 . . 3  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( y  e.  B  /\  x  e.  A )  /\  z  =  C ) }
76tposoprab 6517 . 2  |- tpos  F  =  { <. <. y ,  x >. ,  z >.  |  ( ( y  e.  B  /\  x  e.  A
)  /\  z  =  C ) }
8 df-mpt2 6088 . 2  |-  ( y  e.  B ,  x  e.  A  |->  C )  =  { <. <. y ,  x >. ,  z >.  |  ( ( y  e.  B  /\  x  e.  A )  /\  z  =  C ) }
97, 8eqtr4i 2461 1  |- tpos  F  =  ( y  e.  B ,  x  e.  A  |->  C )
Colors of variables: wff set class
Syntax hints:    /\ wa 360    = wceq 1653    e. wcel 1726   {coprab 6084    e. cmpt2 6085  tpos ctpos 6480
This theorem is referenced by:  oppchomf  13948  oppglsm  15278
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-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-fv 5464  df-oprab 6087  df-mpt2 6088  df-tpos 6481
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