Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fimacnvinrn2 Structured version   Unicode version

Theorem fimacnvinrn2 26099
Description: Taking the converse image of a set can be limited to the range of the function used. (Contributed by Thierry Arnoux, 17-Feb-2017.)
Assertion
Ref Expression
fimacnvinrn2  |-  ( ( Fun  F  /\  ran  F 
C_  B )  -> 
( `' F " A )  =  ( `' F " ( A  i^i  B ) ) )

Proof of Theorem fimacnvinrn2
StepHypRef Expression
1 inass 3663 . . . 4  |-  ( ( A  i^i  B )  i^i  ran  F )  =  ( A  i^i  ( B  i^i  ran  F
) )
2 dfss1 3658 . . . . . . 7  |-  ( ran 
F  C_  B  <->  ( B  i^i  ran  F )  =  ran  F )
32biimpi 194 . . . . . 6  |-  ( ran 
F  C_  B  ->  ( B  i^i  ran  F
)  =  ran  F
)
43adantl 466 . . . . 5  |-  ( ( Fun  F  /\  ran  F 
C_  B )  -> 
( B  i^i  ran  F )  =  ran  F
)
54ineq2d 3655 . . . 4  |-  ( ( Fun  F  /\  ran  F 
C_  B )  -> 
( A  i^i  ( B  i^i  ran  F )
)  =  ( A  i^i  ran  F )
)
61, 5syl5eq 2505 . . 3  |-  ( ( Fun  F  /\  ran  F 
C_  B )  -> 
( ( A  i^i  B )  i^i  ran  F
)  =  ( A  i^i  ran  F )
)
76imaeq2d 5272 . 2  |-  ( ( Fun  F  /\  ran  F 
C_  B )  -> 
( `' F "
( ( A  i^i  B )  i^i  ran  F
) )  =  ( `' F " ( A  i^i  ran  F )
) )
8 fimacnvinrn 26098 . . 3  |-  ( Fun 
F  ->  ( `' F " ( A  i^i  B ) )  =  ( `' F " ( ( A  i^i  B )  i^i  ran  F )
) )
98adantr 465 . 2  |-  ( ( Fun  F  /\  ran  F 
C_  B )  -> 
( `' F "
( A  i^i  B
) )  =  ( `' F " ( ( A  i^i  B )  i^i  ran  F )
) )
10 fimacnvinrn 26098 . . 3  |-  ( Fun 
F  ->  ( `' F " A )  =  ( `' F "
( A  i^i  ran  F ) ) )
1110adantr 465 . 2  |-  ( ( Fun  F  /\  ran  F 
C_  B )  -> 
( `' F " A )  =  ( `' F " ( A  i^i  ran  F )
) )
127, 9, 113eqtr4rd 2504 1  |-  ( ( Fun  F  /\  ran  F 
C_  B )  -> 
( `' F " A )  =  ( `' F " ( A  i^i  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    i^i cin 3430    C_ wss 3431   `'ccnv 4942   ran crn 4944   "cima 4946   Fun wfun 5515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-fo 5527  df-fv 5529
This theorem is referenced by:  eulerpartgbij  26894  orvcval4  26982
  Copyright terms: Public domain W3C validator