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Theorem imaiinfv 28954
Description: Indexed intersection of an image. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Assertion
Ref Expression
imaiinfv  |-  ( ( F  Fn  A  /\  B  C_  A )  ->  |^|_ x  e.  B  ( F `  x )  =  |^| ( F
" B ) )
Distinct variable groups:    x, B    x, F
Allowed substitution hint:    A( x)

Proof of Theorem imaiinfv
StepHypRef Expression
1 fnssres 5521 . . 3  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( F  |`  B )  Fn  B )
2 fniinfv 5747 . . 3  |-  ( ( F  |`  B )  Fn  B  ->  |^|_ x  e.  B  ( ( F  |`  B ) `  x )  =  |^| ran  ( F  |`  B ) )
31, 2syl 16 . 2  |-  ( ( F  Fn  A  /\  B  C_  A )  ->  |^|_ x  e.  B  ( ( F  |`  B ) `
 x )  = 
|^| ran  ( F  |`  B ) )
4 fvres 5701 . . . 4  |-  ( x  e.  B  ->  (
( F  |`  B ) `
 x )  =  ( F `  x
) )
54iineq2i 4187 . . 3  |-  |^|_ x  e.  B  ( ( F  |`  B ) `  x )  =  |^|_ x  e.  B  ( F `
 x )
65eqcomi 2445 . 2  |-  |^|_ x  e.  B  ( F `  x )  =  |^|_ x  e.  B  ( ( F  |`  B ) `  x )
7 df-ima 4849 . . 3  |-  ( F
" B )  =  ran  ( F  |`  B )
87inteqi 4129 . 2  |-  |^| ( F " B )  = 
|^| ran  ( F  |`  B )
93, 6, 83eqtr4g 2498 1  |-  ( ( F  Fn  A  /\  B  C_  A )  ->  |^|_ x  e.  B  ( F `  x )  =  |^| ( F
" B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    C_ wss 3325   |^|cint 4125   |^|_ciin 4169   ran crn 4837    |` cres 4838   "cima 4839    Fn wfn 5410   ` cfv 5415
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 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-int 4126  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-fv 5423
This theorem is referenced by:  elrfirn  28956
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