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Theorem fnresdisj 6688
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 6023 . . 3 Rel (𝐹𝐵)
2 reldm0 5938 . . 3 (Rel (𝐹𝐵) → ((𝐹𝐵) = ∅ ↔ dom (𝐹𝐵) = ∅))
31, 2ax-mp 5 . 2 ((𝐹𝐵) = ∅ ↔ dom (𝐹𝐵) = ∅)
4 dmres 6030 . . . . 5 dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)
5 incom 4209 . . . . 5 (𝐵 ∩ dom 𝐹) = (dom 𝐹𝐵)
64, 5eqtri 2765 . . . 4 dom (𝐹𝐵) = (dom 𝐹𝐵)
7 fndm 6671 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
87ineq1d 4219 . . . 4 (𝐹 Fn 𝐴 → (dom 𝐹𝐵) = (𝐴𝐵))
96, 8eqtrid 2789 . . 3 (𝐹 Fn 𝐴 → dom (𝐹𝐵) = (𝐴𝐵))
109eqeq1d 2739 . 2 (𝐹 Fn 𝐴 → (dom (𝐹𝐵) = ∅ ↔ (𝐴𝐵) = ∅))
113, 10bitr2id 284 1 (𝐹 Fn 𝐴 → ((𝐴𝐵) = ∅ ↔ (𝐹𝐵) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  cin 3950  c0 4333  dom cdm 5685  cres 5687  Rel wrel 5690   Fn wfn 6556
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-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-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-xp 5691  df-rel 5692  df-dm 5695  df-res 5697  df-fn 6564
This theorem is referenced by:  funressn  7179  fvsnun2  7203  dif1enlem  9196  dif1enlemOLD  9197  axdc3lem4  10493  fseq1p1m1  13638  hashgval  14372  hashinf  14374  pwssplit1  21058  mplmonmul  22054  wwlksm1edg  29901  eulerpartlemt  34373  poimirlem3  37630  pwssplit4  43101  isubgr0uhgr  47859
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