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Theorem fnprg 5507
Description: Function with a domain of two different values. (Contributed by FL, 26-Jun-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
fnprg  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  { <. A ,  C >. ,  <. B ,  D >. }  Fn  { A ,  B }
)

Proof of Theorem fnprg
StepHypRef Expression
1 funprg 5502 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  Fun  {
<. A ,  C >. , 
<. B ,  D >. } )
2 dmpropg 5345 . . 3  |-  ( ( C  e.  X  /\  D  e.  Y )  ->  dom  { <. A ,  C >. ,  <. B ,  D >. }  =  { A ,  B }
)
323ad2ant2 980 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  dom  {
<. A ,  C >. , 
<. B ,  D >. }  =  { A ,  B } )
4 df-fn 5459 . 2  |-  ( {
<. A ,  C >. , 
<. B ,  D >. }  Fn  { A ,  B }  <->  ( Fun  { <. A ,  C >. , 
<. B ,  D >. }  /\  dom  { <. A ,  C >. ,  <. B ,  D >. }  =  { A ,  B }
) )
51, 3, 4sylanbrc 647 1  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  { <. A ,  C >. ,  <. B ,  D >. }  Fn  { A ,  B }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   {cpr 3817   <.cop 3819   dom cdm 4880   Fun wfun 5450    Fn wfn 5451
This theorem is referenced by:  f1oprswap  5719  xpscfn  13786  constr3lem4  21636
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-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
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 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-fun 5458  df-fn 5459
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