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Theorem intimass2 36966
 Description: The image under the intersection of relations is a subset of the intersection of the images. (Contributed by RP, 13-Apr-2020.)
Assertion
Ref Expression
intimass2 ( 𝐴𝐵) ⊆ 𝑥𝐴 (𝑥𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem intimass2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 intimass 36965 . 2 ( 𝐴𝐵) ⊆ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝑥𝐵)}
2 intima0 36958 . 2 𝑥𝐴 (𝑥𝐵) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝑥𝐵)}
31, 2sseqtr4i 3601 1 ( 𝐴𝐵) ⊆ 𝑥𝐴 (𝑥𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  {cab 2596  ∃wrex 2897   ⊆ wss 3540  ∩ cint 4410  ∩ ciin 4456   “ 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-8 1979  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  ax-un 6847 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-rex 2902  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-uni 4373  df-int 4411  df-iin 4458  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051 This theorem is referenced by: (None)
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