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Theorem fnresdisj 4952
 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 4582 . . 3 Rel (𝐹B)
2 reldm0 4496 . . 3 (Rel (𝐹B) → ((𝐹B) = ∅ ↔ dom (𝐹B) = ∅))
31, 2ax-mp 7 . 2 ((𝐹B) = ∅ ↔ dom (𝐹B) = ∅)
4 dmres 4575 . . . . 5 dom (𝐹B) = (B ∩ dom 𝐹)
5 incom 3123 . . . . 5 (B ∩ dom 𝐹) = (dom 𝐹B)
64, 5eqtri 2057 . . . 4 dom (𝐹B) = (dom 𝐹B)
7 fndm 4941 . . . . 5 (𝐹 Fn A → dom 𝐹 = A)
87ineq1d 3131 . . . 4 (𝐹 Fn A → (dom 𝐹B) = (AB))
96, 8syl5eq 2081 . . 3 (𝐹 Fn A → dom (𝐹B) = (AB))
109eqeq1d 2045 . 2 (𝐹 Fn A → (dom (𝐹B) = ∅ ↔ (AB) = ∅))
113, 10syl5rbb 182 1 (𝐹 Fn A → ((AB) = ∅ ↔ (𝐹B) = ∅))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242   ∩ cin 2910  ∅c0 3218  dom cdm 4288   ↾ cres 4290  Rel wrel 4293   Fn wfn 4840 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-dm 4298  df-res 4300  df-fn 4848 This theorem is referenced by:  fvsnun2  5304  fseq1p1m1  8726
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