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Theorem funprg 5050
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 967 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  A  e.  V )
2 simp2l 969 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  C  e.  X )
3 funsng 5046 . . . 4  |-  ( ( A  e.  V  /\  C  e.  X )  ->  Fun  { <. A ,  C >. } )
41, 2, 3syl2anc 403 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  Fun  {
<. A ,  C >. } )
5 simp1r 968 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  B  e.  W )
6 simp2r 970 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  D  e.  Y )
7 funsng 5046 . . . 4  |-  ( ( B  e.  W  /\  D  e.  Y )  ->  Fun  { <. B ,  D >. } )
85, 6, 7syl2anc 403 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  Fun  {
<. B ,  D >. } )
9 dmsnopg 4889 . . . . . 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 4889 . . . . . 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 3200 . . . 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 3500 . . . . 5  |-  ( A  =/=  B  ->  ( { A }  i^i  { B } )  =  (/) )
15143ad2ant3 966 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  ( { A }  i^i  { B } )  =  (/) )
1613, 15eqtrd 2120 . . 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 5044 . . 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 1173 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  Fun  ( { <. A ,  C >. }  u.  { <. B ,  D >. } ) )
19 df-pr 3448 . . 3  |-  { <. A ,  C >. ,  <. B ,  D >. }  =  ( { <. A ,  C >. }  u.  { <. B ,  D >. } )
2019funeqi 5022 . 2  |-  ( Fun 
{ <. A ,  C >. ,  <. B ,  D >. }  <->  Fun  ( { <. A ,  C >. }  u.  {
<. B ,  D >. } ) )
2118, 20sylibr 132 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 102    /\ w3a 924    = wceq 1289    e. wcel 1438    =/= wne 2255    u. cun 2995    i^i cin 2996   (/)c0 3284   {csn 3441   {cpr 3442   <.cop 3444   dom cdm 4428   Fun wfun 4996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-fun 5004
This theorem is referenced by:  funtpg  5051  funpr  5052  fnprg  5055
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