Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  imarnf1pr Structured version   Unicode version

Theorem imarnf1pr 32683
Description: The image of the range of a function  F under a function  E if  F is a function of a pair into the domain of  E. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
Assertion
Ref Expression
imarnf1pr  |-  ( ( X  e.  V  /\  Y  e.  W )  ->  ( ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( ( E `
 ( F `  X ) )  =  A  /\  ( E `
 ( F `  Y ) )  =  B ) )  -> 
( E " ran  F )  =  { A ,  B } ) )

Proof of Theorem imarnf1pr
StepHypRef Expression
1 ffn 5713 . . . . . . . . 9  |-  ( E : dom  E --> R  ->  E  Fn  dom  E )
21adantl 464 . . . . . . . 8  |-  ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  ->  E  Fn  dom  E )
32adantr 463 . . . . . . 7  |-  ( ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( X  e.  V  /\  Y  e.  W
) )  ->  E  Fn  dom  E )
4 simpll 751 . . . . . . . 8  |-  ( ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( X  e.  V  /\  Y  e.  W
) )  ->  F : { X ,  Y }
--> dom  E )
5 prid1g 4122 . . . . . . . . . 10  |-  ( X  e.  V  ->  X  e.  { X ,  Y } )
65adantr 463 . . . . . . . . 9  |-  ( ( X  e.  V  /\  Y  e.  W )  ->  X  e.  { X ,  Y } )
76adantl 464 . . . . . . . 8  |-  ( ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( X  e.  V  /\  Y  e.  W
) )  ->  X  e.  { X ,  Y } )
84, 7ffvelrnd 6008 . . . . . . 7  |-  ( ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( X  e.  V  /\  Y  e.  W
) )  ->  ( F `  X )  e.  dom  E )
9 prid2g 4123 . . . . . . . . 9  |-  ( Y  e.  W  ->  Y  e.  { X ,  Y } )
109ad2antll 726 . . . . . . . 8  |-  ( ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( X  e.  V  /\  Y  e.  W
) )  ->  Y  e.  { X ,  Y } )
114, 10ffvelrnd 6008 . . . . . . 7  |-  ( ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( X  e.  V  /\  Y  e.  W
) )  ->  ( F `  Y )  e.  dom  E )
12 fnimapr 5912 . . . . . . 7  |-  ( ( E  Fn  dom  E  /\  ( F `  X
)  e.  dom  E  /\  ( F `  Y
)  e.  dom  E
)  ->  ( E " { ( F `  X ) ,  ( F `  Y ) } )  =  {
( E `  ( F `  X )
) ,  ( E `
 ( F `  Y ) ) } )
133, 8, 11, 12syl3anc 1226 . . . . . 6  |-  ( ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( X  e.  V  /\  Y  e.  W
) )  ->  ( E " { ( F `
 X ) ,  ( F `  Y
) } )  =  { ( E `  ( F `  X ) ) ,  ( E `
 ( F `  Y ) ) } )
1413ex 432 . . . . 5  |-  ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  ->  (
( X  e.  V  /\  Y  e.  W
)  ->  ( E " { ( F `  X ) ,  ( F `  Y ) } )  =  {
( E `  ( F `  X )
) ,  ( E `
 ( F `  Y ) ) } ) )
1514adantr 463 . . . 4  |-  ( ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( ( E `  ( F `  X ) )  =  A  /\  ( E `  ( F `
 Y ) )  =  B ) )  ->  ( ( X  e.  V  /\  Y  e.  W )  ->  ( E " { ( F `
 X ) ,  ( F `  Y
) } )  =  { ( E `  ( F `  X ) ) ,  ( E `
 ( F `  Y ) ) } ) )
1615impcom 428 . . 3  |-  ( ( ( X  e.  V  /\  Y  e.  W
)  /\  ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( ( E `
 ( F `  X ) )  =  A  /\  ( E `
 ( F `  Y ) )  =  B ) ) )  ->  ( E " { ( F `  X ) ,  ( F `  Y ) } )  =  {
( E `  ( F `  X )
) ,  ( E `
 ( F `  Y ) ) } )
17 ffn 5713 . . . . . . . . 9  |-  ( F : { X ,  Y } --> dom  E  ->  F  Fn  { X ,  Y } )
18 rnfdmpr 32682 . . . . . . . . 9  |-  ( ( X  e.  V  /\  Y  e.  W )  ->  ( F  Fn  { X ,  Y }  ->  ran  F  =  {
( F `  X
) ,  ( F `
 Y ) } ) )
1917, 18syl5com 30 . . . . . . . 8  |-  ( F : { X ,  Y } --> dom  E  ->  ( ( X  e.  V  /\  Y  e.  W
)  ->  ran  F  =  { ( F `  X ) ,  ( F `  Y ) } ) )
2019adantr 463 . . . . . . 7  |-  ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  ->  (
( X  e.  V  /\  Y  e.  W
)  ->  ran  F  =  { ( F `  X ) ,  ( F `  Y ) } ) )
2120adantr 463 . . . . . 6  |-  ( ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( ( E `  ( F `  X ) )  =  A  /\  ( E `  ( F `
 Y ) )  =  B ) )  ->  ( ( X  e.  V  /\  Y  e.  W )  ->  ran  F  =  { ( F `
 X ) ,  ( F `  Y
) } ) )
2221impcom 428 . . . . 5  |-  ( ( ( X  e.  V  /\  Y  e.  W
)  /\  ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( ( E `
 ( F `  X ) )  =  A  /\  ( E `
 ( F `  Y ) )  =  B ) ) )  ->  ran  F  =  { ( F `  X ) ,  ( F `  Y ) } )
2322eqcomd 2462 . . . 4  |-  ( ( ( X  e.  V  /\  Y  e.  W
)  /\  ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( ( E `
 ( F `  X ) )  =  A  /\  ( E `
 ( F `  Y ) )  =  B ) ) )  ->  { ( F `
 X ) ,  ( F `  Y
) }  =  ran  F )
2423imaeq2d 5325 . . 3  |-  ( ( ( X  e.  V  /\  Y  e.  W
)  /\  ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( ( E `
 ( F `  X ) )  =  A  /\  ( E `
 ( F `  Y ) )  =  B ) ) )  ->  ( E " { ( F `  X ) ,  ( F `  Y ) } )  =  ( E " ran  F
) )
25 preq12 4097 . . . 4  |-  ( ( ( E `  ( F `  X )
)  =  A  /\  ( E `  ( F `
 Y ) )  =  B )  ->  { ( E `  ( F `  X ) ) ,  ( E `
 ( F `  Y ) ) }  =  { A ,  B } )
2625ad2antll 726 . . 3  |-  ( ( ( X  e.  V  /\  Y  e.  W
)  /\  ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( ( E `
 ( F `  X ) )  =  A  /\  ( E `
 ( F `  Y ) )  =  B ) ) )  ->  { ( E `
 ( F `  X ) ) ,  ( E `  ( F `  Y )
) }  =  { A ,  B }
)
2716, 24, 263eqtr3d 2503 . 2  |-  ( ( ( X  e.  V  /\  Y  e.  W
)  /\  ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( ( E `
 ( F `  X ) )  =  A  /\  ( E `
 ( F `  Y ) )  =  B ) ) )  ->  ( E " ran  F )  =  { A ,  B }
)
2827ex 432 1  |-  ( ( X  e.  V  /\  Y  e.  W )  ->  ( ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( ( E `
 ( F `  X ) )  =  A  /\  ( E `
 ( F `  Y ) )  =  B ) )  -> 
( E " ran  F )  =  { A ,  B } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   {cpr 4018   dom cdm 4988   ran crn 4989   "cima 4991    Fn wfn 5565   -->wf 5566   ` cfv 5570
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-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-sbc 3325  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-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator