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Theorem fnimadisj 5910
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 5889 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
21ineq1d 3774 . . . 4 (𝐹 Fn 𝐴 → (dom 𝐹𝐶) = (𝐴𝐶))
32eqeq1d 2611 . . 3 (𝐹 Fn 𝐴 → ((dom 𝐹𝐶) = ∅ ↔ (𝐴𝐶) = ∅))
43biimpar 500 . 2 ((𝐹 Fn 𝐴 ∧ (𝐴𝐶) = ∅) → (dom 𝐹𝐶) = ∅)
5 imadisj 5389 . 2 ((𝐹𝐶) = ∅ ↔ (dom 𝐹𝐶) = ∅)
64, 5sylibr 222 1 ((𝐹 Fn 𝐴 ∧ (𝐴𝐶) = ∅) → (𝐹𝐶) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  cin 3538  c0 3873  dom cdm 5027  cima 5030   Fn wfn 5784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-xp 5033  df-cnv 5035  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-fn 5792
This theorem is referenced by:  poimirlem15  32377  aacllem  42298
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