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Theorem fnresdisj 6162
Description: A function restricted to a class disjoint with its domain is empty. (Contributed by NM, 23-Sep-2004.)
Assertion
Ref Expression
fnresdisj (𝐹 Fn 𝐴 → ((𝐴𝐵) = ∅ ↔ (𝐹𝐵) = ∅))

Proof of Theorem fnresdisj
StepHypRef Expression
1 relres 5584 . . 3 Rel (𝐹𝐵)
2 reldm0 5498 . . 3 (Rel (𝐹𝐵) → ((𝐹𝐵) = ∅ ↔ dom (𝐹𝐵) = ∅))
31, 2ax-mp 5 . 2 ((𝐹𝐵) = ∅ ↔ dom (𝐹𝐵) = ∅)
4 dmres 5577 . . . . 5 dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)
5 incom 3948 . . . . 5 (𝐵 ∩ dom 𝐹) = (dom 𝐹𝐵)
64, 5eqtri 2782 . . . 4 dom (𝐹𝐵) = (dom 𝐹𝐵)
7 fndm 6151 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
87ineq1d 3956 . . . 4 (𝐹 Fn 𝐴 → (dom 𝐹𝐵) = (𝐴𝐵))
96, 8syl5eq 2806 . . 3 (𝐹 Fn 𝐴 → dom (𝐹𝐵) = (𝐴𝐵))
109eqeq1d 2762 . 2 (𝐹 Fn 𝐴 → (dom (𝐹𝐵) = ∅ ↔ (𝐴𝐵) = ∅))
113, 10syl5rbb 273 1 (𝐹 Fn 𝐴 → ((𝐴𝐵) = ∅ ↔ (𝐹𝐵) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1632  cin 3714  c0 4058  dom cdm 5266  cres 5268  Rel wrel 5271   Fn wfn 6044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-dm 5276  df-res 5278  df-fn 6052
This theorem is referenced by:  funressn  6590  fvsnun2  6614  axdc3lem4  9487  fseq1p1m1  12627  hashgval  13334  hashinf  13336  pwssplit1  19281  mplmonmul  19686  wwlksm1edg  27011  eulerpartlemt  30763  poimirlem3  33743  pwssplit4  38179
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