MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funimass1 Structured version   Visualization version   GIF version

Theorem funimass1 6648
Description: A kind of contraposition law that infers a subclass of an image from a preimage subclass. (Contributed by NM, 25-May-2004.)
Assertion
Ref Expression
funimass1 ((Fun 𝐹𝐴 ⊆ ran 𝐹) → ((𝐹𝐴) ⊆ 𝐵𝐴 ⊆ (𝐹𝐵)))

Proof of Theorem funimass1
StepHypRef Expression
1 imass2 6120 . 2 ((𝐹𝐴) ⊆ 𝐵 → (𝐹 “ (𝐹𝐴)) ⊆ (𝐹𝐵))
2 funimacnv 6647 . . . 4 (Fun 𝐹 → (𝐹 “ (𝐹𝐴)) = (𝐴 ∩ ran 𝐹))
3 dfss 3970 . . . . . 6 (𝐴 ⊆ ran 𝐹𝐴 = (𝐴 ∩ ran 𝐹))
43biimpi 216 . . . . 5 (𝐴 ⊆ ran 𝐹𝐴 = (𝐴 ∩ ran 𝐹))
54eqcomd 2743 . . . 4 (𝐴 ⊆ ran 𝐹 → (𝐴 ∩ ran 𝐹) = 𝐴)
62, 5sylan9eq 2797 . . 3 ((Fun 𝐹𝐴 ⊆ ran 𝐹) → (𝐹 “ (𝐹𝐴)) = 𝐴)
76sseq1d 4015 . 2 ((Fun 𝐹𝐴 ⊆ ran 𝐹) → ((𝐹 “ (𝐹𝐴)) ⊆ (𝐹𝐵) ↔ 𝐴 ⊆ (𝐹𝐵)))
81, 7imbitrid 244 1 ((Fun 𝐹𝐴 ⊆ ran 𝐹) → ((𝐹𝐴) ⊆ 𝐵𝐴 ⊆ (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  cin 3950  wss 3951  ccnv 5684  ran crn 5686  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-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-sb 2065  df-mo 2540  df-eu 2569  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:  kqnrmlem1  23751  hmeontr  23777  nrmhmph  23802  cnheiborlem  24986
  Copyright terms: Public domain W3C validator