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Theorem fnprg 5305
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 5301 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  Fun  {
<. A ,  C >. , 
<. B ,  D >. } )
2 dmpropg 5146 . . 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 5258 . 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 1623    e. wcel 1684    =/= wne 2446   {cpr 3641   <.cop 3643   dom cdm 4689   Fun wfun 5249    Fn wfn 5250
This theorem is referenced by:  f1oprswap  5515  xpscfn  13461  repcpwti  25161
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-fun 5257  df-fn 5258
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