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Theorem intima0 36233
Description: Two ways of expressing the intersection of images of a class. (Contributed by RP, 13-Apr-2020.)
Assertion
Ref Expression
intima0  |-  |^|_ a  e.  A  ( a " B )  =  |^| { x  |  E. a  e.  A  x  =  ( a " B
) }
Distinct variable groups:    x, A    x, B    x, a
Allowed substitution hints:    A( a)    B( a)

Proof of Theorem intima0
StepHypRef Expression
1 vex 3047 . . 3  |-  a  e. 
_V
2 imaexg 6727 . . 3  |-  ( a  e.  _V  ->  (
a " B )  e.  _V )
31, 2ax-mp 5 . 2  |-  ( a
" B )  e. 
_V
43dfiin2 4312 1  |-  |^|_ a  e.  A  ( a " B )  =  |^| { x  |  E. a  e.  A  x  =  ( a " B
) }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1443    e. wcel 1886   {cab 2436   E.wrex 2737   _Vcvv 3044   |^|cint 4233   |^|_ciin 4278   "cima 4836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-int 4234  df-iin 4280  df-br 4402  df-opab 4461  df-xp 4839  df-cnv 4841  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846
This theorem is referenced by:  intimass2  36241  intimasn2  36244
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