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Theorem fnresdisj 6618
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 5964 . . 3 Rel (𝐹𝐵)
2 reldm0 5881 . . 3 (Rel (𝐹𝐵) → ((𝐹𝐵) = ∅ ↔ dom (𝐹𝐵) = ∅))
31, 2ax-mp 5 . 2 ((𝐹𝐵) = ∅ ↔ dom (𝐹𝐵) = ∅)
4 dmres 5957 . . . . 5 dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)
5 incom 4159 . . . . 5 (𝐵 ∩ dom 𝐹) = (dom 𝐹𝐵)
64, 5eqtri 2764 . . . 4 dom (𝐹𝐵) = (dom 𝐹𝐵)
7 fndm 6602 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
87ineq1d 4169 . . . 4 (𝐹 Fn 𝐴 → (dom 𝐹𝐵) = (𝐴𝐵))
96, 8eqtrid 2788 . . 3 (𝐹 Fn 𝐴 → dom (𝐹𝐵) = (𝐴𝐵))
109eqeq1d 2738 . 2 (𝐹 Fn 𝐴 → (dom (𝐹𝐵) = ∅ ↔ (𝐴𝐵) = ∅))
113, 10bitr2id 283 1 (𝐹 Fn 𝐴 → ((𝐴𝐵) = ∅ ↔ (𝐹𝐵) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  cin 3907  c0 4280  dom cdm 5631  cres 5633  Rel wrel 5636   Fn wfn 6488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5166  df-xp 5637  df-rel 5638  df-dm 5641  df-res 5643  df-fn 6496
This theorem is referenced by:  funressn  7101  fvsnun2  7125  dif1enlem  9096  dif1enlemOLD  9097  axdc3lem4  10385  fseq1p1m1  13507  hashgval  14225  hashinf  14227  pwssplit1  20505  mplmonmul  21421  wwlksm1edg  28712  eulerpartlemt  32840  poimirlem3  36048  pwssplit4  41354
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