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Theorem f1oprg 5721
Description: An unordered pair of ordered pairs with different elements is a one-to-one onto function, analogous to f1oprswap 5720. (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 5719 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  { <. A ,  B >. } : { A }
-1-1-onto-> { B } )
21ad2antrr 708 . . . 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 5719 . . . . 5  |-  ( ( C  e.  X  /\  D  e.  Y )  ->  { <. C ,  D >. } : { C }
-1-1-onto-> { D } )
43ad2antlr 709 . . . 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 3871 . . . . 5  |-  ( A  =/=  C  ->  ( { A }  i^i  { C } )  =  (/) )
65ad2antrl 710 . . . 4  |-  ( ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )
)  /\  ( A  =/=  C  /\  B  =/= 
D ) )  -> 
( { A }  i^i  { C } )  =  (/) )
7 disjsn2 3871 . . . . 5  |-  ( B  =/=  D  ->  ( { B }  i^i  { D } )  =  (/) )
87ad2antll 711 . . . 4  |-  ( ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )
)  /\  ( A  =/=  C  /\  B  =/= 
D ) )  -> 
( { B }  i^i  { D } )  =  (/) )
9 f1oun 5697 . . . 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 1186 . . 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 3823 . . . . . 6  |-  { <. A ,  B >. ,  <. C ,  D >. }  =  ( { <. A ,  B >. }  u.  { <. C ,  D >. } )
1211eqcomi 2442 . . . . 5  |-  ( {
<. A ,  B >. }  u.  { <. C ,  D >. } )  =  { <. A ,  B >. ,  <. C ,  D >. }
1312a1i 11 . . . 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 3823 . . . . . 6  |-  { A ,  C }  =  ( { A }  u.  { C } )
1514eqcomi 2442 . . . . 5  |-  ( { A }  u.  { C } )  =  { A ,  C }
1615a1i 11 . . . 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 3823 . . . . . 6  |-  { B ,  D }  =  ( { B }  u.  { D } )
1817eqcomi 2442 . . . . 5  |-  ( { B }  u.  { D } )  =  { B ,  D }
1918a1i 11 . . . 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 5674 . . 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 203 . 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 425 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 360    = wceq 1653    e. wcel 1726    =/= wne 2601    u. cun 3320    i^i cin 3321   (/)c0 3630   {csn 3816   {cpr 3817   <.cop 3819   -1-1-onto->wf1o 5456
This theorem is referenced by:  s2f1o  11868  f1oun2prg  11869  2trllemE  21558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
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-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464
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