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Theorem intima0 36093
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 3081 . . 3  |-  a  e. 
_V
2 imaexg 6736 . . 3  |-  ( a  e.  _V  ->  (
a " B )  e.  _V )
31, 2ax-mp 5 . 2  |-  ( a
" B )  e. 
_V
43dfiin2 4328 1  |-  |^|_ a  e.  A  ( a " B )  =  |^| { x  |  E. a  e.  A  x  =  ( a " B
) }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437    e. wcel 1867   {cab 2405   E.wrex 2774   _Vcvv 3078   |^|cint 4249   |^|_ciin 4294   "cima 4849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4540  ax-nul 4548  ax-pr 4653  ax-un 6589
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-int 4250  df-iin 4296  df-br 4418  df-opab 4477  df-xp 4852  df-cnv 4854  df-dm 4856  df-rn 4857  df-res 4858  df-ima 4859
This theorem is referenced by:  intimass2  36101  intimasn2  36104
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