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Theorem inimass 5468
 Description: The image of an intersection. (Contributed by Thierry Arnoux, 16-Dec-2017.)
Assertion
Ref Expression
inimass ((𝐴𝐵) “ 𝐶) ⊆ ((𝐴𝐶) ∩ (𝐵𝐶))

Proof of Theorem inimass
StepHypRef Expression
1 rnin 5461 . 2 ran ((𝐴𝐶) ∩ (𝐵𝐶)) ⊆ (ran (𝐴𝐶) ∩ ran (𝐵𝐶))
2 df-ima 5051 . . 3 ((𝐴𝐵) “ 𝐶) = ran ((𝐴𝐵) ↾ 𝐶)
3 resindir 5333 . . . 4 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))
43rneqi 5273 . . 3 ran ((𝐴𝐵) ↾ 𝐶) = ran ((𝐴𝐶) ∩ (𝐵𝐶))
52, 4eqtri 2632 . 2 ((𝐴𝐵) “ 𝐶) = ran ((𝐴𝐶) ∩ (𝐵𝐶))
6 df-ima 5051 . . 3 (𝐴𝐶) = ran (𝐴𝐶)
7 df-ima 5051 . . 3 (𝐵𝐶) = ran (𝐵𝐶)
86, 7ineq12i 3774 . 2 ((𝐴𝐶) ∩ (𝐵𝐶)) = (ran (𝐴𝐶) ∩ ran (𝐵𝐶))
91, 5, 83sstr4i 3607 1 ((𝐴𝐵) “ 𝐶) ⊆ ((𝐴𝐶) ∩ (𝐵𝐶))
 Colors of variables: wff setvar class Syntax hints:   ∩ cin 3539   ⊆ wss 3540  ran crn 5039   ↾ cres 5040   “ cima 5041 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051 This theorem is referenced by:  restutopopn  21852
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