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Theorem intpreima 5656
Description: Preimage of an intersection. (Contributed by FL, 28-Apr-2012.)
Assertion
Ref Expression
intpreima  |-  ( ( Fun  F  /\  A  =/=  (/) )  ->  ( `' F " |^| A
)  =  |^|_ x  e.  A  ( `' F " x ) )
Distinct variable groups:    x, A    x, F

Proof of Theorem intpreima
StepHypRef Expression
1 intiin 3956 . . 3  |-  |^| A  =  |^|_ x  e.  A  x
21imaeq2i 5010 . 2  |-  ( `' F " |^| A
)  =  ( `' F " |^|_ x  e.  A  x )
3 iinpreima 5655 . 2  |-  ( ( Fun  F  /\  A  =/=  (/) )  ->  ( `' F " |^|_ x  e.  A  x )  =  |^|_ x  e.  A  ( `' F " x ) )
42, 3syl5eq 2327 1  |-  ( ( Fun  F  /\  A  =/=  (/) )  ->  ( `' F " |^| A
)  =  |^|_ x  e.  A  ( `' F " x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    =/= wne 2446   (/)c0 3455   |^|cint 3862   |^|_ciin 3906   `'ccnv 4688   "cima 4692   Fun wfun 5249
This theorem is referenced by:  subbascn  16984
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-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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