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Theorem funtpg 5620
Description: A set of three pairs is a function if their first members are different. (Contributed by Alexander van der Vekens, 5-Dec-2017.)
Assertion
Ref Expression
funtpg  |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W
)  /\  ( A  e.  F  /\  B  e.  G  /\  C  e.  H )  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/= 
Z ) )  ->  Fun  { <. X ,  A >. ,  <. Y ,  B >. ,  <. Z ,  C >. } )

Proof of Theorem funtpg
StepHypRef Expression
1 3simpa 991 . . . 4  |-  ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W )  ->  ( X  e.  U  /\  Y  e.  V
) )
2 3simpa 991 . . . 4  |-  ( ( A  e.  F  /\  B  e.  G  /\  C  e.  H )  ->  ( A  e.  F  /\  B  e.  G
) )
3 simp1 994 . . . 4  |-  ( ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z )  ->  X  =/=  Y )
4 funprg 5619 . . . 4  |-  ( ( ( X  e.  U  /\  Y  e.  V
)  /\  ( A  e.  F  /\  B  e.  G )  /\  X  =/=  Y )  ->  Fun  {
<. X ,  A >. , 
<. Y ,  B >. } )
51, 2, 3, 4syl3an 1268 . . 3  |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W
)  /\  ( A  e.  F  /\  B  e.  G  /\  C  e.  H )  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/= 
Z ) )  ->  Fun  { <. X ,  A >. ,  <. Y ,  B >. } )
6 simp13 1026 . . . 4  |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W
)  /\  ( A  e.  F  /\  B  e.  G  /\  C  e.  H )  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/= 
Z ) )  ->  Z  e.  W )
7 simp23 1029 . . . 4  |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W
)  /\  ( A  e.  F  /\  B  e.  G  /\  C  e.  H )  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/= 
Z ) )  ->  C  e.  H )
8 funsng 5616 . . . 4  |-  ( ( Z  e.  W  /\  C  e.  H )  ->  Fun  { <. Z ,  C >. } )
96, 7, 8syl2anc 659 . . 3  |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W
)  /\  ( A  e.  F  /\  B  e.  G  /\  C  e.  H )  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/= 
Z ) )  ->  Fun  { <. Z ,  C >. } )
1023ad2ant2 1016 . . . . . 6  |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W
)  /\  ( A  e.  F  /\  B  e.  G  /\  C  e.  H )  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/= 
Z ) )  -> 
( A  e.  F  /\  B  e.  G
) )
11 dmpropg 5464 . . . . . 6  |-  ( ( A  e.  F  /\  B  e.  G )  ->  dom  { <. X ,  A >. ,  <. Y ,  B >. }  =  { X ,  Y }
)
1210, 11syl 16 . . . . 5  |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W
)  /\  ( A  e.  F  /\  B  e.  G  /\  C  e.  H )  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/= 
Z ) )  ->  dom  { <. X ,  A >. ,  <. Y ,  B >. }  =  { X ,  Y } )
13 dmsnopg 5462 . . . . . 6  |-  ( C  e.  H  ->  dom  {
<. Z ,  C >. }  =  { Z }
)
147, 13syl 16 . . . . 5  |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W
)  /\  ( A  e.  F  /\  B  e.  G  /\  C  e.  H )  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/= 
Z ) )  ->  dom  { <. Z ,  C >. }  =  { Z } )
1512, 14ineq12d 3687 . . . 4  |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W
)  /\  ( A  e.  F  /\  B  e.  G  /\  C  e.  H )  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/= 
Z ) )  -> 
( dom  { <. X ,  A >. ,  <. Y ,  B >. }  i^i  dom  {
<. Z ,  C >. } )  =  ( { X ,  Y }  i^i  { Z } ) )
16 elpri 4036 . . . . . . . 8  |-  ( Z  e.  { X ,  Y }  ->  ( Z  =  X  \/  Z  =  Y ) )
17 nne 2655 . . . . . . . . . . . . 13  |-  ( -.  X  =/=  Z  <->  X  =  Z )
1817biimpri 206 . . . . . . . . . . . 12  |-  ( X  =  Z  ->  -.  X  =/=  Z )
1918eqcoms 2466 . . . . . . . . . . 11  |-  ( Z  =  X  ->  -.  X  =/=  Z )
20193mix2d 1170 . . . . . . . . . 10  |-  ( Z  =  X  ->  ( -.  X  =/=  Y  \/  -.  X  =/=  Z  \/  -.  Y  =/=  Z
) )
21 nne 2655 . . . . . . . . . . . . 13  |-  ( -.  Y  =/=  Z  <->  Y  =  Z )
2221biimpri 206 . . . . . . . . . . . 12  |-  ( Y  =  Z  ->  -.  Y  =/=  Z )
2322eqcoms 2466 . . . . . . . . . . 11  |-  ( Z  =  Y  ->  -.  Y  =/=  Z )
24233mix3d 1171 . . . . . . . . . 10  |-  ( Z  =  Y  ->  ( -.  X  =/=  Y  \/  -.  X  =/=  Z  \/  -.  Y  =/=  Z
) )
2520, 24jaoi 377 . . . . . . . . 9  |-  ( ( Z  =  X  \/  Z  =  Y )  ->  ( -.  X  =/= 
Y  \/  -.  X  =/=  Z  \/  -.  Y  =/=  Z ) )
26 3ianor 988 . . . . . . . . 9  |-  ( -.  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z
)  <->  ( -.  X  =/=  Y  \/  -.  X  =/=  Z  \/  -.  Y  =/=  Z ) )
2725, 26sylibr 212 . . . . . . . 8  |-  ( ( Z  =  X  \/  Z  =  Y )  ->  -.  ( X  =/= 
Y  /\  X  =/=  Z  /\  Y  =/=  Z
) )
2816, 27syl 16 . . . . . . 7  |-  ( Z  e.  { X ,  Y }  ->  -.  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/= 
Z ) )
2928con2i 120 . . . . . 6  |-  ( ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z )  ->  -.  Z  e.  { X ,  Y } )
30 disjsn 4076 . . . . . 6  |-  ( ( { X ,  Y }  i^i  { Z }
)  =  (/)  <->  -.  Z  e.  { X ,  Y } )
3129, 30sylibr 212 . . . . 5  |-  ( ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z )  ->  ( { X ,  Y }  i^i  { Z } )  =  (/) )
32313ad2ant3 1017 . . . 4  |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W
)  /\  ( A  e.  F  /\  B  e.  G  /\  C  e.  H )  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/= 
Z ) )  -> 
( { X ,  Y }  i^i  { Z } )  =  (/) )
3315, 32eqtrd 2495 . . 3  |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W
)  /\  ( A  e.  F  /\  B  e.  G  /\  C  e.  H )  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/= 
Z ) )  -> 
( dom  { <. X ,  A >. ,  <. Y ,  B >. }  i^i  dom  {
<. Z ,  C >. } )  =  (/) )
34 funun 5612 . . 3  |-  ( ( ( Fun  { <. X ,  A >. ,  <. Y ,  B >. }  /\  Fun  { <. Z ,  C >. } )  /\  ( dom  { <. X ,  A >. ,  <. Y ,  B >. }  i^i  dom  { <. Z ,  C >. } )  =  (/) )  ->  Fun  ( { <. X ,  A >. ,  <. Y ,  B >. }  u.  { <. Z ,  C >. } ) )
355, 9, 33, 34syl21anc 1225 . 2  |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W
)  /\  ( A  e.  F  /\  B  e.  G  /\  C  e.  H )  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/= 
Z ) )  ->  Fun  ( { <. X ,  A >. ,  <. Y ,  B >. }  u.  { <. Z ,  C >. } ) )
36 df-tp 4021 . . 3  |-  { <. X ,  A >. ,  <. Y ,  B >. ,  <. Z ,  C >. }  =  ( { <. X ,  A >. ,  <. Y ,  B >. }  u.  { <. Z ,  C >. } )
3736funeqi 5590 . 2  |-  ( Fun 
{ <. X ,  A >. ,  <. Y ,  B >. ,  <. Z ,  C >. }  <->  Fun  ( { <. X ,  A >. ,  <. Y ,  B >. }  u.  {
<. Z ,  C >. } ) )
3835, 37sylibr 212 1  |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W
)  /\  ( A  e.  F  /\  B  e.  G  /\  C  e.  H )  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/= 
Z ) )  ->  Fun  { <. X ,  A >. ,  <. Y ,  B >. ,  <. Z ,  C >. } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 366    /\ wa 367    \/ w3o 970    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649    u. cun 3459    i^i cin 3460   (/)c0 3783   {csn 4016   {cpr 4018   {ctp 4020   <.cop 4022   dom cdm 4988   Fun wfun 5564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-fun 5572
This theorem is referenced by:  fntpg  5625  estrres  15607  constr2spthlem1  24798
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