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Theorem fnresdisj 6039
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 5461 . . 3 Rel (𝐹𝐵)
2 reldm0 5375 . . 3 (Rel (𝐹𝐵) → ((𝐹𝐵) = ∅ ↔ dom (𝐹𝐵) = ∅))
31, 2ax-mp 5 . 2 ((𝐹𝐵) = ∅ ↔ dom (𝐹𝐵) = ∅)
4 dmres 5454 . . . . 5 dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)
5 incom 3838 . . . . 5 (𝐵 ∩ dom 𝐹) = (dom 𝐹𝐵)
64, 5eqtri 2673 . . . 4 dom (𝐹𝐵) = (dom 𝐹𝐵)
7 fndm 6028 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
87ineq1d 3846 . . . 4 (𝐹 Fn 𝐴 → (dom 𝐹𝐵) = (𝐴𝐵))
96, 8syl5eq 2697 . . 3 (𝐹 Fn 𝐴 → dom (𝐹𝐵) = (𝐴𝐵))
109eqeq1d 2653 . 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 1523  cin 3606  c0 3948  dom cdm 5143  cres 5145  Rel wrel 5148   Fn wfn 5921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-dm 5153  df-res 5155  df-fn 5929
This theorem is referenced by:  funressn  6466  fvsnun2  6490  axdc3lem4  9313  fseq1p1m1  12452  hashgval  13160  hashinf  13162  pwssplit1  19107  mplmonmul  19512  wwlksm1edg  26835  eulerpartlemt  30561  poimirlem3  33542  pwssplit4  37976
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