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Theorem f1oprg 28185
Description: An unordered pair of ordered pairs with different elements is a one-to-one onto function, analogous to f1oprswap 5531. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
Assertion
Ref Expression
f1oprg  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y ) )  -> 
( ( A  =/= 
C  /\  B  =/=  D )  ->  { <. A ,  B >. ,  <. C ,  D >. } : { A ,  C } -1-1-onto-> { B ,  D }
) )

Proof of Theorem f1oprg
StepHypRef Expression
1 f1osng 5530 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  { <. A ,  B >. } : { A }
-1-1-onto-> { B } )
21ad2antrr 706 . . . 4  |-  ( ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )
)  /\  ( A  =/=  C  /\  B  =/= 
D ) )  ->  { <. A ,  B >. } : { A }
-1-1-onto-> { B } )
3 f1osng 5530 . . . . 5  |-  ( ( C  e.  X  /\  D  e.  Y )  ->  { <. C ,  D >. } : { C }
-1-1-onto-> { D } )
43ad2antlr 707 . . . 4  |-  ( ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )
)  /\  ( A  =/=  C  /\  B  =/= 
D ) )  ->  { <. C ,  D >. } : { C }
-1-1-onto-> { D } )
5 disjsn2 3707 . . . . 5  |-  ( A  =/=  C  ->  ( { A }  i^i  { C } )  =  (/) )
65ad2antrl 708 . . . 4  |-  ( ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )
)  /\  ( A  =/=  C  /\  B  =/= 
D ) )  -> 
( { A }  i^i  { C } )  =  (/) )
7 disjsn2 3707 . . . . 5  |-  ( B  =/=  D  ->  ( { B }  i^i  { D } )  =  (/) )
87ad2antll 709 . . . 4  |-  ( ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )
)  /\  ( A  =/=  C  /\  B  =/= 
D ) )  -> 
( { B }  i^i  { D } )  =  (/) )
9 f1oun 5508 . . . 4  |-  ( ( ( { <. A ,  B >. } : { A } -1-1-onto-> { B }  /\  {
<. C ,  D >. } : { C } -1-1-onto-> { D } )  /\  (
( { A }  i^i  { C } )  =  (/)  /\  ( { B }  i^i  { D } )  =  (/) ) )  ->  ( { <. A ,  B >. }  u.  { <. C ,  D >. } ) : ( { A }  u.  { C } ) -1-1-onto-> ( { B }  u.  { D } ) )
102, 4, 6, 8, 9syl22anc 1183 . . 3  |-  ( ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )
)  /\  ( A  =/=  C  /\  B  =/= 
D ) )  -> 
( { <. A ,  B >. }  u.  { <. C ,  D >. } ) : ( { A }  u.  { C } ) -1-1-onto-> ( { B }  u.  { D } ) )
11 df-pr 3660 . . . . . 6  |-  { <. A ,  B >. ,  <. C ,  D >. }  =  ( { <. A ,  B >. }  u.  { <. C ,  D >. } )
1211eqcomi 2300 . . . . 5  |-  ( {
<. A ,  B >. }  u.  { <. C ,  D >. } )  =  { <. A ,  B >. ,  <. C ,  D >. }
1312a1i 10 . . . 4  |-  ( ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )
)  /\  ( A  =/=  C  /\  B  =/= 
D ) )  -> 
( { <. A ,  B >. }  u.  { <. C ,  D >. } )  =  { <. A ,  B >. ,  <. C ,  D >. } )
14 df-pr 3660 . . . . . 6  |-  { A ,  C }  =  ( { A }  u.  { C } )
1514eqcomi 2300 . . . . 5  |-  ( { A }  u.  { C } )  =  { A ,  C }
1615a1i 10 . . . 4  |-  ( ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )
)  /\  ( A  =/=  C  /\  B  =/= 
D ) )  -> 
( { A }  u.  { C } )  =  { A ,  C } )
17 df-pr 3660 . . . . . 6  |-  { B ,  D }  =  ( { B }  u.  { D } )
1817eqcomi 2300 . . . . 5  |-  ( { B }  u.  { D } )  =  { B ,  D }
1918a1i 10 . . . 4  |-  ( ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )
)  /\  ( A  =/=  C  /\  B  =/= 
D ) )  -> 
( { B }  u.  { D } )  =  { B ,  D } )
2013, 16, 19f1oeq123d 5485 . . 3  |-  ( ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )
)  /\  ( A  =/=  C  /\  B  =/= 
D ) )  -> 
( ( { <. A ,  B >. }  u.  {
<. C ,  D >. } ) : ( { A }  u.  { C } ) -1-1-onto-> ( { B }  u.  { D } )  <->  { <. A ,  B >. ,  <. C ,  D >. } : { A ,  C } -1-1-onto-> { B ,  D } ) )
2110, 20mpbid 201 . 2  |-  ( ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )
)  /\  ( A  =/=  C  /\  B  =/= 
D ) )  ->  { <. A ,  B >. ,  <. C ,  D >. } : { A ,  C } -1-1-onto-> { B ,  D } )
2221ex 423 1  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y ) )  -> 
( ( A  =/= 
C  /\  B  =/=  D )  ->  { <. A ,  B >. ,  <. C ,  D >. } : { A ,  C } -1-1-onto-> { B ,  D }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459    u. cun 3163    i^i cin 3164   (/)c0 3468   {csn 3653   {cpr 3654   <.cop 3656   -1-1-onto->wf1o 5270
This theorem is referenced by:  f1oun2prg  28186  s2f1o  28222
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278
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