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Theorem inimass 5360
Description: The image of an intersection (Contributed by Thierry Arnoux, 16-Dec-2017.)
Assertion
Ref Expression
inimass  |-  ( ( A  i^i  B )
" C )  C_  ( ( A " C )  i^i  ( B " C ) )

Proof of Theorem inimass
StepHypRef Expression
1 rnin 5353 . 2  |-  ran  (
( A  |`  C )  i^i  ( B  |`  C ) )  C_  ( ran  ( A  |`  C )  i^i  ran  ( B  |`  C ) )
2 df-ima 4960 . . 3  |-  ( ( A  i^i  B )
" C )  =  ran  ( ( A  i^i  B )  |`  C )
3 resindir 5234 . . . 4  |-  ( ( A  i^i  B )  |`  C )  =  ( ( A  |`  C )  i^i  ( B  |`  C ) )
43rneqi 5173 . . 3  |-  ran  (
( A  i^i  B
)  |`  C )  =  ran  ( ( A  |`  C )  i^i  ( B  |`  C ) )
52, 4eqtri 2483 . 2  |-  ( ( A  i^i  B )
" C )  =  ran  ( ( A  |`  C )  i^i  ( B  |`  C ) )
6 df-ima 4960 . . 3  |-  ( A
" C )  =  ran  ( A  |`  C )
7 df-ima 4960 . . 3  |-  ( B
" C )  =  ran  ( B  |`  C )
86, 7ineq12i 3657 . 2  |-  ( ( A " C )  i^i  ( B " C ) )  =  ( ran  ( A  |`  C )  i^i  ran  ( B  |`  C ) )
91, 5, 83sstr4i 3502 1  |-  ( ( A  i^i  B )
" C )  C_  ( ( A " C )  i^i  ( B " C ) )
Colors of variables: wff setvar class
Syntax hints:    i^i cin 3434    C_ wss 3435   ran crn 4948    |` cres 4949   "cima 4950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pr 4638
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-br 4400  df-opab 4458  df-xp 4953  df-rel 4954  df-cnv 4955  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960
This theorem is referenced by:  restutopopn  19944
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