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Theorem funprg 5630
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 1015 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  A  e.  V )
2 simp2l 1017 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  C  e.  X )
3 funsng 5627 . . . 4  |-  ( ( A  e.  V  /\  C  e.  X )  ->  Fun  { <. A ,  C >. } )
41, 2, 3syl2anc 661 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  Fun  {
<. A ,  C >. } )
5 simp1r 1016 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  B  e.  W )
6 simp2r 1018 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  D  e.  Y )
7 funsng 5627 . . . 4  |-  ( ( B  e.  W  /\  D  e.  Y )  ->  Fun  { <. B ,  D >. } )
85, 6, 7syl2anc 661 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  Fun  {
<. B ,  D >. } )
9 dmsnopg 5472 . . . . . 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 5472 . . . . . 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 3696 . . . 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 4084 . . . . 5  |-  ( A  =/=  B  ->  ( { A }  i^i  { B } )  =  (/) )
15143ad2ant3 1014 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  ( { A }  i^i  { B } )  =  (/) )
1613, 15eqtrd 2503 . . 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 5623 . . 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 1222 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  Fun  ( { <. A ,  C >. }  u.  { <. B ,  D >. } ) )
19 df-pr 4025 . . 3  |-  { <. A ,  C >. ,  <. B ,  D >. }  =  ( { <. A ,  C >. }  u.  { <. B ,  D >. } )
2019funeqi 5601 . 2  |-  ( Fun 
{ <. A ,  C >. ,  <. B ,  D >. }  <->  Fun  ( { <. A ,  C >. }  u.  {
<. B ,  D >. } ) )
2118, 20sylibr 212 1  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B )  ->  Fun  {
<. A ,  C >. , 
<. B ,  D >. } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2657    u. cun 3469    i^i cin 3470   (/)c0 3780   {csn 4022   {cpr 4024   <.cop 4028   dom cdm 4994   Fun wfun 5575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-br 4443  df-opab 4501  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-fun 5583
This theorem is referenced by:  funtpg  5631  funpr  5632  fnprg  5635  constr3pthlem2  24320
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