MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funprg Structured version   Unicode version

Theorem funprg 5502
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 982 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  A  e.  V )
2 simp2l 984 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  C  e.  X )
3 funsng 5499 . . . 4  |-  ( ( A  e.  V  /\  C  e.  X )  ->  Fun  { <. A ,  C >. } )
41, 2, 3syl2anc 644 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  Fun  {
<. A ,  C >. } )
5 simp1r 983 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  B  e.  W )
6 simp2r 985 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  D  e.  Y )
7 funsng 5499 . . . 4  |-  ( ( B  e.  W  /\  D  e.  Y )  ->  Fun  { <. B ,  D >. } )
85, 6, 7syl2anc 644 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  Fun  {
<. B ,  D >. } )
9 dmsnopg 5343 . . . . . 6  |-  ( C  e.  X  ->  dom  {
<. A ,  C >. }  =  { A }
)
102, 9syl 16 . . . . 5  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  dom  {
<. A ,  C >. }  =  { A }
)
11 dmsnopg 5343 . . . . . 6  |-  ( D  e.  Y  ->  dom  {
<. B ,  D >. }  =  { B }
)
126, 11syl 16 . . . . 5  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  dom  {
<. B ,  D >. }  =  { B }
)
1310, 12ineq12d 3545 . . . 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 3871 . . . . 5  |-  ( A  =/=  B  ->  ( { A }  i^i  { B } )  =  (/) )
15143ad2ant3 981 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  ( { A }  i^i  { B } )  =  (/) )
1613, 15eqtrd 2470 . . 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 5497 . . 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 1184 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  Fun  ( { <. A ,  C >. }  u.  { <. B ,  D >. } ) )
19 df-pr 3823 . . 3  |-  { <. A ,  C >. ,  <. B ,  D >. }  =  ( { <. A ,  C >. }  u.  { <. B ,  D >. } )
2019funeqi 5476 . 2  |-  ( Fun 
{ <. A ,  C >. ,  <. B ,  D >. }  <->  Fun  ( { <. A ,  C >. }  u.  {
<. B ,  D >. } ) )
2118, 20sylibr 205 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601    u. cun 3320    i^i cin 3321   (/)c0 3630   {csn 3816   {cpr 3817   <.cop 3819   dom cdm 4880   Fun wfun 5450
This theorem is referenced by:  funtpg  5503  funpr  5504  fnprg  5507  constr3pthlem2  21645
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-fun 5458
  Copyright terms: Public domain W3C validator