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Theorem inimass 4923
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 4916 . 2 ran ((𝐴𝐶) ∩ (𝐵𝐶)) ⊆ (ran (𝐴𝐶) ∩ ran (𝐵𝐶))
2 df-ima 4520 . . 3 ((𝐴𝐵) “ 𝐶) = ran ((𝐴𝐵) ↾ 𝐶)
3 resindir 4803 . . . 4 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))
43rneqi 4735 . . 3 ran ((𝐴𝐵) ↾ 𝐶) = ran ((𝐴𝐶) ∩ (𝐵𝐶))
52, 4eqtri 2136 . 2 ((𝐴𝐵) “ 𝐶) = ran ((𝐴𝐶) ∩ (𝐵𝐶))
6 df-ima 4520 . . 3 (𝐴𝐶) = ran (𝐴𝐶)
7 df-ima 4520 . . 3 (𝐵𝐶) = ran (𝐵𝐶)
86, 7ineq12i 3243 . 2 ((𝐴𝐶) ∩ (𝐵𝐶)) = (ran (𝐴𝐶) ∩ ran (𝐵𝐶))
91, 5, 83sstr4i 3106 1 ((𝐴𝐵) “ 𝐶) ⊆ ((𝐴𝐶) ∩ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  cin 3038  wss 3039  ran crn 4508  cres 4509  cima 4510
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-xp 4513  df-rel 4514  df-cnv 4515  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520
This theorem is referenced by: (None)
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