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Theorem imaiinfv 26432
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 5499 . . 3  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( F  |`  B )  Fn  B )
2 fniinfv 5725 . . 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 5686 . . . 4  |-  ( x  e.  B  ->  (
( F  |`  B ) `
 x )  =  ( F `  x
) )
54iineq2i 4055 . . 3  |-  |^|_ x  e.  B  ( ( F  |`  B ) `  x )  =  |^|_ x  e.  B  ( F `
 x )
65eqcomi 2392 . 2  |-  |^|_ x  e.  B  ( F `  x )  =  |^|_ x  e.  B  ( ( F  |`  B ) `  x )
7 df-ima 4832 . . 3  |-  ( F
" B )  =  ran  ( F  |`  B )
87inteqi 3997 . 2  |-  |^| ( F " B )  = 
|^| ran  ( F  |`  B )
93, 6, 83eqtr4g 2445 1  |-  ( ( F  Fn  A  /\  B  C_  A )  ->  |^|_ x  e.  B  ( F `  x )  =  |^| ( F
" B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    C_ wss 3264   |^|cint 3993   |^|_ciin 4037   ran crn 4820    |` cres 4821   "cima 4822    Fn wfn 5390   ` cfv 5395
This theorem is referenced by:  elrfirn  26441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-int 3994  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-fv 5403
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