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Theorem inimass 5453
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 5446 . 2 ran ((𝐴𝐶) ∩ (𝐵𝐶)) ⊆ (ran (𝐴𝐶) ∩ ran (𝐵𝐶))
2 df-ima 5040 . . 3 ((𝐴𝐵) “ 𝐶) = ran ((𝐴𝐵) ↾ 𝐶)
3 resindir 5319 . . . 4 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))
43rneqi 5259 . . 3 ran ((𝐴𝐵) ↾ 𝐶) = ran ((𝐴𝐶) ∩ (𝐵𝐶))
52, 4eqtri 2631 . 2 ((𝐴𝐵) “ 𝐶) = ran ((𝐴𝐶) ∩ (𝐵𝐶))
6 df-ima 5040 . . 3 (𝐴𝐶) = ran (𝐴𝐶)
7 df-ima 5040 . . 3 (𝐵𝐶) = ran (𝐵𝐶)
86, 7ineq12i 3773 . 2 ((𝐴𝐶) ∩ (𝐵𝐶)) = (ran (𝐴𝐶) ∩ ran (𝐵𝐶))
91, 5, 83sstr4i 3606 1 ((𝐴𝐵) “ 𝐶) ⊆ ((𝐴𝐶) ∩ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  cin 3538  wss 3539  ran crn 5028  cres 5029  cima 5030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-xp 5033  df-rel 5034  df-cnv 5035  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040
This theorem is referenced by:  restutopopn  21799
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