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Theorem imaiinfv 26759
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 5357 . . 3  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( F  |`  B )  Fn  B )
2 fniinfv 5581 . . 3  |-  ( ( F  |`  B )  Fn  B  ->  |^|_ x  e.  B  ( ( F  |`  B ) `  x )  =  |^| ran  ( F  |`  B ) )
31, 2syl 15 . 2  |-  ( ( F  Fn  A  /\  B  C_  A )  ->  |^|_ x  e.  B  ( ( F  |`  B ) `
 x )  = 
|^| ran  ( F  |`  B ) )
4 fvres 5542 . . . 4  |-  ( x  e.  B  ->  (
( F  |`  B ) `
 x )  =  ( F `  x
) )
54iineq2i 3924 . . 3  |-  |^|_ x  e.  B  ( ( F  |`  B ) `  x )  =  |^|_ x  e.  B  ( F `
 x )
65eqcomi 2287 . 2  |-  |^|_ x  e.  B  ( F `  x )  =  |^|_ x  e.  B  ( ( F  |`  B ) `  x )
7 df-ima 4702 . . 3  |-  ( F
" B )  =  ran  ( F  |`  B )
87inteqi 3866 . 2  |-  |^| ( F " B )  = 
|^| ran  ( F  |`  B )
93, 6, 83eqtr4g 2340 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 358    = wceq 1623    C_ wss 3152   |^|cint 3862   |^|_ciin 3906   ran crn 4690    |` cres 4691   "cima 4692    Fn wfn 5250   ` cfv 5255
This theorem is referenced by:  elrfirn  26770
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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