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Theorem intpreima 6010
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 4379 . . 3  |-  |^| A  =  |^|_ x  e.  A  x
21imaeq2i 5333 . 2  |-  ( `' F " |^| A
)  =  ( `' F " |^|_ x  e.  A  x )
3 iinpreima 6009 . 2  |-  ( ( Fun  F  /\  A  =/=  (/) )  ->  ( `' F " |^|_ x  e.  A  x )  =  |^|_ x  e.  A  ( `' F " x ) )
42, 3syl5eq 2520 1  |-  ( ( Fun  F  /\  A  =/=  (/) )  ->  ( `' F " |^| A
)  =  |^|_ x  e.  A  ( `' F " x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    =/= wne 2662   (/)c0 3785   |^|cint 4282   |^|_ciin 4326   `'ccnv 4998   "cima 5002   Fun wfun 5580
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-iin 4328  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-fv 5594
This theorem is referenced by:  subbascn  19521
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