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Theorem intimass2 38789
 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 38788 . 2 ( 𝐴𝐵) ⊆ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝑥𝐵)}
2 intima0 38781 . 2 𝑥𝐴 (𝑥𝐵) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝑥𝐵)}
31, 2sseqtr4i 3864 1 ( 𝐴𝐵) ⊆ 𝑥𝐴 (𝑥𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1658  {cab 2812  ∃wrex 3119   ⊆ wss 3799  ∩ cint 4698  ∩ ciin 4742   “ cima 5346 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128  ax-un 7210 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-int 4699  df-iin 4744  df-br 4875  df-opab 4937  df-xp 5349  df-cnv 5351  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356 This theorem is referenced by: (None)
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