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Theorem intpreima 5998
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 4327 . . 3  |-  |^| A  =  |^|_ x  e.  A  x
21imaeq2i 5157 . 2  |-  ( `' F " |^| A
)  =  ( `' F " |^|_ x  e.  A  x )
3 iinpreima 5997 . 2  |-  ( ( Fun  F  /\  A  =/=  (/) )  ->  ( `' F " |^|_ x  e.  A  x )  =  |^|_ x  e.  A  ( `' F " x ) )
42, 3syl5eq 2457 1  |-  ( ( Fun  F  /\  A  =/=  (/) )  ->  ( `' F " |^| A
)  =  |^|_ x  e.  A  ( `' F " x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1407    =/= wne 2600   (/)c0 3740   |^|cint 4229   |^|_ciin 4274   `'ccnv 4824   "cima 4828   Fun wfun 5565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-int 4230  df-iin 4276  df-br 4398  df-opab 4456  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-fv 5579
This theorem is referenced by:  subbascn  20050
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