MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tposmpt2 Unicode version

Theorem tposmpt2 6271
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 5863 . . . 4  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
3 ancom 437 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  B )  <->  ( y  e.  B  /\  x  e.  A )
)
43anbi1i 676 . . . . 5  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  z  =  C )  <->  ( (
y  e.  B  /\  x  e.  A )  /\  z  =  C
) )
54oprabbii 5903 . . . 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 2307 . . 3  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( y  e.  B  /\  x  e.  A )  /\  z  =  C ) }
76tposoprab 6270 . 2  |- tpos  F  =  { <. <. y ,  x >. ,  z >.  |  ( ( y  e.  B  /\  x  e.  A
)  /\  z  =  C ) }
8 df-mpt2 5863 . 2  |-  ( y  e.  B ,  x  e.  A  |->  C )  =  { <. <. y ,  x >. ,  z >.  |  ( ( y  e.  B  /\  x  e.  A )  /\  z  =  C ) }
97, 8eqtr4i 2306 1  |- tpos  F  =  ( y  e.  B ,  x  e.  A  |->  C )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684   {coprab 5859    e. cmpt2 5860  tpos ctpos 6233
This theorem is referenced by:  oppchomf  13623  oppglsm  14953
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-oprab 5862  df-mpt2 5863  df-tpos 6234
  Copyright terms: Public domain W3C validator