MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  intpreima Structured version   Visualization version   Unicode version

Theorem intpreima 6026
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 4323 . . 3  |-  |^| A  =  |^|_ x  e.  A  x
21imaeq2i 5172 . 2  |-  ( `' F " |^| A
)  =  ( `' F " |^|_ x  e.  A  x )
3 iinpreima 6025 . 2  |-  ( ( Fun  F  /\  A  =/=  (/) )  ->  ( `' F " |^|_ x  e.  A  x )  =  |^|_ x  e.  A  ( `' F " x ) )
42, 3syl5eq 2517 1  |-  ( ( Fun  F  /\  A  =/=  (/) )  ->  ( `' F " |^| A
)  =  |^|_ x  e.  A  ( `' F " x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    =/= wne 2641   (/)c0 3722   |^|cint 4226   |^|_ciin 4270   `'ccnv 4838   "cima 4842   Fun wfun 5583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-int 4227  df-iin 4272  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-fv 5597
This theorem is referenced by:  subbascn  20347
  Copyright terms: Public domain W3C validator