MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sspreima Structured version   Visualization version   GIF version

Theorem sspreima 7088
Description: The preimage of a subset is a subset of the preimage. (Contributed by Brendan Leahy, 23-Sep-2017.)
Assertion
Ref Expression
sspreima ((Fun 𝐹𝐴𝐵) → (𝐹𝐴) ⊆ (𝐹𝐵))

Proof of Theorem sspreima
StepHypRef Expression
1 inpreima 7084 . . 3 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∩ (𝐹𝐵)))
2 dfss2 3969 . . . . 5 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
32biimpi 216 . . . 4 (𝐴𝐵 → (𝐴𝐵) = 𝐴)
43imaeq2d 6078 . . 3 (𝐴𝐵 → (𝐹 “ (𝐴𝐵)) = (𝐹𝐴))
51, 4sylan9req 2798 . 2 ((Fun 𝐹𝐴𝐵) → ((𝐹𝐴) ∩ (𝐹𝐵)) = (𝐹𝐴))
6 dfss2 3969 . 2 ((𝐹𝐴) ⊆ (𝐹𝐵) ↔ ((𝐹𝐴) ∩ (𝐹𝐵)) = (𝐹𝐴))
75, 6sylibr 234 1 ((Fun 𝐹𝐴𝐵) → (𝐹𝐴) ⊆ (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  cin 3950  wss 3951  ccnv 5684  cima 5688  Fun wfun 6555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563
This theorem is referenced by:  pwrssmgc  32990  gsumhashmul  33064  elrspunidl  33456  carsggect  34320  eulerpartlemmf  34377  eulerpartlemgf  34381  orvclteinc  34478  cnneiima  48814
  Copyright terms: Public domain W3C validator