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Theorem funprg 5143
Description: A set of two pairs is a function if their first members are different. (Contributed by FL, 26-Jun-2011.)
Assertion
Ref Expression
funprg  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  Fun  {
<. A ,  C >. , 
<. B ,  D >. } )

Proof of Theorem funprg
StepHypRef Expression
1 simp1l 990 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  A  e.  V )
2 simp2l 992 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  C  e.  X )
3 funsng 5139 . . . 4  |-  ( ( A  e.  V  /\  C  e.  X )  ->  Fun  { <. A ,  C >. } )
41, 2, 3syl2anc 408 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  Fun  {
<. A ,  C >. } )
5 simp1r 991 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  B  e.  W )
6 simp2r 993 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  D  e.  Y )
7 funsng 5139 . . . 4  |-  ( ( B  e.  W  /\  D  e.  Y )  ->  Fun  { <. B ,  D >. } )
85, 6, 7syl2anc 408 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  Fun  {
<. B ,  D >. } )
9 dmsnopg 4980 . . . . . 6  |-  ( C  e.  X  ->  dom  {
<. A ,  C >. }  =  { A }
)
102, 9syl 14 . . . . 5  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  dom  {
<. A ,  C >. }  =  { A }
)
11 dmsnopg 4980 . . . . . 6  |-  ( D  e.  Y  ->  dom  {
<. B ,  D >. }  =  { B }
)
126, 11syl 14 . . . . 5  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  dom  {
<. B ,  D >. }  =  { B }
)
1310, 12ineq12d 3248 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  ( dom  { <. A ,  C >. }  i^i  dom  { <. B ,  D >. } )  =  ( { A }  i^i  { B } ) )
14 disjsn2 3556 . . . . 5  |-  ( A  =/=  B  ->  ( { A }  i^i  { B } )  =  (/) )
15143ad2ant3 989 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  ( { A }  i^i  { B } )  =  (/) )
1613, 15eqtrd 2150 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  ( dom  { <. A ,  C >. }  i^i  dom  { <. B ,  D >. } )  =  (/) )
17 funun 5137 . . 3  |-  ( ( ( Fun  { <. A ,  C >. }  /\  Fun  { <. B ,  D >. } )  /\  ( dom  { <. A ,  C >. }  i^i  dom  { <. B ,  D >. } )  =  (/) )  ->  Fun  ( { <. A ,  C >. }  u.  { <. B ,  D >. } ) )
184, 8, 16, 17syl21anc 1200 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  Fun  ( { <. A ,  C >. }  u.  { <. B ,  D >. } ) )
19 df-pr 3504 . . 3  |-  { <. A ,  C >. ,  <. B ,  D >. }  =  ( { <. A ,  C >. }  u.  { <. B ,  D >. } )
2019funeqi 5114 . 2  |-  ( Fun 
{ <. A ,  C >. ,  <. B ,  D >. }  <->  Fun  ( { <. A ,  C >. }  u.  {
<. B ,  D >. } ) )
2118, 20sylibr 133 1  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  Fun  {
<. A ,  C >. , 
<. B ,  D >. } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 947    = wceq 1316    e. wcel 1465    =/= wne 2285    u. cun 3039    i^i cin 3040   (/)c0 3333   {csn 3497   {cpr 3498   <.cop 3500   dom cdm 4509   Fun wfun 5087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-fun 5095
This theorem is referenced by:  funtpg  5144  funpr  5145  fnprg  5148  2strbasg  11987  2stropg  11988
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