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Theorem fimacnvinrn2 27697
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 3694 . . . 4  |-  ( ( A  i^i  B )  i^i  ran  F )  =  ( A  i^i  ( B  i^i  ran  F
) )
2 dfss1 3689 . . . . . . 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 464 . . . . 5  |-  ( ( Fun  F  /\  ran  F 
C_  B )  -> 
( B  i^i  ran  F )  =  ran  F
)
54ineq2d 3686 . . . 4  |-  ( ( Fun  F  /\  ran  F 
C_  B )  -> 
( A  i^i  ( B  i^i  ran  F )
)  =  ( A  i^i  ran  F )
)
61, 5syl5eq 2507 . . 3  |-  ( ( Fun  F  /\  ran  F 
C_  B )  -> 
( ( A  i^i  B )  i^i  ran  F
)  =  ( A  i^i  ran  F )
)
76imaeq2d 5325 . 2  |-  ( ( Fun  F  /\  ran  F 
C_  B )  -> 
( `' F "
( ( A  i^i  B )  i^i  ran  F
) )  =  ( `' F " ( A  i^i  ran  F )
) )
8 fimacnvinrn 27696 . . 3  |-  ( Fun 
F  ->  ( `' F " ( A  i^i  B ) )  =  ( `' F " ( ( A  i^i  B )  i^i  ran  F )
) )
98adantr 463 . 2  |-  ( ( Fun  F  /\  ran  F 
C_  B )  -> 
( `' F "
( A  i^i  B
) )  =  ( `' F " ( ( A  i^i  B )  i^i  ran  F )
) )
10 fimacnvinrn 27696 . . 3  |-  ( Fun 
F  ->  ( `' F " A )  =  ( `' F "
( A  i^i  ran  F ) ) )
1110adantr 463 . 2  |-  ( ( Fun  F  /\  ran  F 
C_  B )  -> 
( `' F " A )  =  ( `' F " ( A  i^i  ran  F )
) )
127, 9, 113eqtr4rd 2506 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 367    = wceq 1398    i^i cin 3460    C_ wss 3461   `'ccnv 4987   ran crn 4989   "cima 4991   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-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-br 4440  df-opab 4498  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-fo 5576  df-fv 5578
This theorem is referenced by:  eulerpartgbij  28575  orvcval4  28663
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