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Theorem f1oprg 5677
Description: An unordered pair of ordered pairs with different elements is a one-to-one onto function, analogous to f1oprswap 5676. (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 5675 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  { <. A ,  B >. } : { A }
-1-1-onto-> { B } )
21ad2antrr 707 . . . 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 5675 . . . . 5  |-  ( ( C  e.  X  /\  D  e.  Y )  ->  { <. C ,  D >. } : { C }
-1-1-onto-> { D } )
43ad2antlr 708 . . . 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 3829 . . . . 5  |-  ( A  =/=  C  ->  ( { A }  i^i  { C } )  =  (/) )
65ad2antrl 709 . . . 4  |-  ( ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )
)  /\  ( A  =/=  C  /\  B  =/= 
D ) )  -> 
( { A }  i^i  { C } )  =  (/) )
7 disjsn2 3829 . . . . 5  |-  ( B  =/=  D  ->  ( { B }  i^i  { D } )  =  (/) )
87ad2antll 710 . . . 4  |-  ( ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )
)  /\  ( A  =/=  C  /\  B  =/= 
D ) )  -> 
( { B }  i^i  { D } )  =  (/) )
9 f1oun 5653 . . . 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 1185 . . 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 3781 . . . . . 6  |-  { <. A ,  B >. ,  <. C ,  D >. }  =  ( { <. A ,  B >. }  u.  { <. C ,  D >. } )
1211eqcomi 2408 . . . . 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 3781 . . . . . 6  |-  { A ,  C }  =  ( { A }  u.  { C } )
1514eqcomi 2408 . . . . 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 3781 . . . . . 6  |-  { B ,  D }  =  ( { B }  u.  { D } )
1817eqcomi 2408 . . . . 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 5630 . . 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 202 . 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 424 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 359    = wceq 1649    e. wcel 1721    =/= wne 2567    u. cun 3278    i^i cin 3279   (/)c0 3588   {csn 3774   {cpr 3775   <.cop 3777   -1-1-onto->wf1o 5412
This theorem is referenced by:  s2f1o  11818  f1oun2prg  11819  2trllemE  21506
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420
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