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

Theorem fnimadisj 6151
Description: A class that is disjoint with the domain of a function has an empty image under the function. (Contributed by FL, 24-Jan-2007.)
Assertion
Ref Expression
fnimadisj ((𝐹 Fn 𝐴 ∧ (𝐴𝐶) = ∅) → (𝐹𝐶) = ∅)

Proof of Theorem fnimadisj
StepHypRef Expression
1 fndm 6129 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
21ineq1d 3964 . . . 4 (𝐹 Fn 𝐴 → (dom 𝐹𝐶) = (𝐴𝐶))
32eqeq1d 2773 . . 3 (𝐹 Fn 𝐴 → ((dom 𝐹𝐶) = ∅ ↔ (𝐴𝐶) = ∅))
43biimpar 463 . 2 ((𝐹 Fn 𝐴 ∧ (𝐴𝐶) = ∅) → (dom 𝐹𝐶) = ∅)
5 imadisj 5624 . 2 ((𝐹𝐶) = ∅ ↔ (dom 𝐹𝐶) = ∅)
64, 5sylibr 224 1 ((𝐹 Fn 𝐴 ∧ (𝐴𝐶) = ∅) → (𝐹𝐶) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  cin 3722  c0 4063  dom cdm 5250  cima 5253   Fn wfn 6025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-xp 5256  df-cnv 5258  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-fn 6033
This theorem is referenced by:  poimirlem15  33757  aacllem  43075
  Copyright terms: Public domain W3C validator