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Theorem funprg 5262
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 1021 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  A  e.  V )
2 simp2l 1023 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  C  e.  X )
3 funsng 5258 . . . 4  |-  ( ( A  e.  V  /\  C  e.  X )  ->  Fun  { <. A ,  C >. } )
41, 2, 3syl2anc 411 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  Fun  {
<. A ,  C >. } )
5 simp1r 1022 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  B  e.  W )
6 simp2r 1024 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  D  e.  Y )
7 funsng 5258 . . . 4  |-  ( ( B  e.  W  /\  D  e.  Y )  ->  Fun  { <. B ,  D >. } )
85, 6, 7syl2anc 411 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  Fun  {
<. B ,  D >. } )
9 dmsnopg 5096 . . . . . 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 5096 . . . . . 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 3337 . . . 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 3654 . . . . 5  |-  ( A  =/=  B  ->  ( { A }  i^i  { B } )  =  (/) )
15143ad2ant3 1020 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  ( { A }  i^i  { B } )  =  (/) )
1613, 15eqtrd 2210 . . 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 5256 . . 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 1237 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  Fun  ( { <. A ,  C >. }  u.  { <. B ,  D >. } ) )
19 df-pr 3598 . . 3  |-  { <. A ,  C >. ,  <. B ,  D >. }  =  ( { <. A ,  C >. }  u.  { <. B ,  D >. } )
2019funeqi 5233 . 2  |-  ( Fun 
{ <. A ,  C >. ,  <. B ,  D >. }  <->  Fun  ( { <. A ,  C >. }  u.  {
<. B ,  D >. } ) )
2118, 20sylibr 134 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 104    /\ w3a 978    = wceq 1353    e. wcel 2148    =/= wne 2347    u. cun 3127    i^i cin 3128   (/)c0 3422   {csn 3591   {cpr 3592   <.cop 3594   dom cdm 4623   Fun wfun 5206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-fun 5214
This theorem is referenced by:  funtpg  5263  funpr  5264  fnprg  5267  2strbasg  12554  2stropg  12555
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