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Theorem inimass 4925
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 4918 . 2 ran ((𝐴𝐶) ∩ (𝐵𝐶)) ⊆ (ran (𝐴𝐶) ∩ ran (𝐵𝐶))
2 df-ima 4522 . . 3 ((𝐴𝐵) “ 𝐶) = ran ((𝐴𝐵) ↾ 𝐶)
3 resindir 4805 . . . 4 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))
43rneqi 4737 . . 3 ran ((𝐴𝐵) ↾ 𝐶) = ran ((𝐴𝐶) ∩ (𝐵𝐶))
52, 4eqtri 2138 . 2 ((𝐴𝐵) “ 𝐶) = ran ((𝐴𝐶) ∩ (𝐵𝐶))
6 df-ima 4522 . . 3 (𝐴𝐶) = ran (𝐴𝐶)
7 df-ima 4522 . . 3 (𝐵𝐶) = ran (𝐵𝐶)
86, 7ineq12i 3245 . 2 ((𝐴𝐶) ∩ (𝐵𝐶)) = (ran (𝐴𝐶) ∩ ran (𝐵𝐶))
91, 5, 83sstr4i 3108 1 ((𝐴𝐵) “ 𝐶) ⊆ ((𝐴𝐶) ∩ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  cin 3040  wss 3041  ran crn 4510  cres 4511  cima 4512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-rel 4516  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522
This theorem is referenced by: (None)
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