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Theorem fnresdisj 6606
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 5958 . . 3 Rel (𝐹𝐵)
2 reldm0 5872 . . 3 (Rel (𝐹𝐵) → ((𝐹𝐵) = ∅ ↔ dom (𝐹𝐵) = ∅))
31, 2ax-mp 5 . 2 ((𝐹𝐵) = ∅ ↔ dom (𝐹𝐵) = ∅)
4 dmres 5965 . . . . 5 dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)
5 incom 4158 . . . . 5 (𝐵 ∩ dom 𝐹) = (dom 𝐹𝐵)
64, 5eqtri 2756 . . . 4 dom (𝐹𝐵) = (dom 𝐹𝐵)
7 fndm 6589 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
87ineq1d 4168 . . . 4 (𝐹 Fn 𝐴 → (dom 𝐹𝐵) = (𝐴𝐵))
96, 8eqtrid 2780 . . 3 (𝐹 Fn 𝐴 → dom (𝐹𝐵) = (𝐴𝐵))
109eqeq1d 2735 . 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 1541  cin 3897  c0 4282  dom cdm 5619  cres 5621  Rel wrel 5624   Fn wfn 6481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-xp 5625  df-rel 5626  df-dm 5629  df-res 5631  df-fn 6489
This theorem is referenced by:  funressn  7098  fvsnun2  7123  dif1enlem  9076  axdc3lem4  10351  fseq1p1m1  13500  hashgval  14242  hashinf  14244  pwssplit1  20995  mplmonmul  21972  wwlksm1edg  29861  eulerpartlemt  34405  poimirlem3  37683  pwssplit4  43206  isubgr0uhgr  47997
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