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Theorem intpreima 5834
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 4224 . . 3  |-  |^| A  =  |^|_ x  e.  A  x
21imaeq2i 5167 . 2  |-  ( `' F " |^| A
)  =  ( `' F " |^|_ x  e.  A  x )
3 iinpreima 5833 . 2  |-  ( ( Fun  F  /\  A  =/=  (/) )  ->  ( `' F " |^|_ x  e.  A  x )  =  |^|_ x  e.  A  ( `' F " x ) )
42, 3syl5eq 2487 1  |-  ( ( Fun  F  /\  A  =/=  (/) )  ->  ( `' F " |^| A
)  =  |^|_ x  e.  A  ( `' F " x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    =/= wne 2606   (/)c0 3637   |^|cint 4128   |^|_ciin 4172   `'ccnv 4839   "cima 4843   Fun wfun 5412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-int 4129  df-iin 4174  df-br 4293  df-opab 4351  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-fv 5426
This theorem is referenced by:  subbascn  18858
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