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Theorem fnprg 5321
Description: Domain of a 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 5317 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  Fun  {
<. A ,  C >. , 
<. B ,  D >. } )
2 dmpropg 5162 . . 3  |-  ( ( C  e.  X  /\  D  e.  Y )  ->  dom  { <. A ,  C >. ,  <. B ,  D >. }  =  { A ,  B }
)
323ad2ant2 977 . 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 5274 . 2  |-  ( {
<. A ,  C >. , 
<. B ,  D >. }  Fn  { A ,  B }  <->  ( Fun  { <. A ,  C >. , 
<. B ,  D >. }  /\  dom  { <. A ,  C >. ,  <. B ,  D >. }  =  { A ,  B }
) )
51, 3, 4sylanbrc 645 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   {cpr 3654   <.cop 3656   dom cdm 4705   Fun wfun 5265    Fn wfn 5266
This theorem is referenced by:  f1oprswap  5531  xpscfn  13477  repcpwti  25264  constr3lem4  28393
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-fun 5273  df-fn 5274
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