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Theorem ndmima 4645
 Description: The image of a singleton outside the domain is empty. (Contributed by NM, 22-May-1998.)
Assertion
Ref Expression
ndmima A dom B → (B “ {A}) = ∅)

Proof of Theorem ndmima
StepHypRef Expression
1 df-ima 4301 . 2 (B “ {A}) = ran (B ↾ {A})
2 dmres 4575 . . . . 5 dom (B ↾ {A}) = ({A} ∩ dom B)
3 incom 3123 . . . . 5 ({A} ∩ dom B) = (dom B ∩ {A})
42, 3eqtri 2057 . . . 4 dom (B ↾ {A}) = (dom B ∩ {A})
5 disjsn 3423 . . . . 5 ((dom B ∩ {A}) = ∅ ↔ ¬ A dom B)
65biimpri 124 . . . 4 A dom B → (dom B ∩ {A}) = ∅)
74, 6syl5eq 2081 . . 3 A dom B → dom (B ↾ {A}) = ∅)
8 dm0rn0 4495 . . 3 (dom (B ↾ {A}) = ∅ ↔ ran (B ↾ {A}) = ∅)
97, 8sylib 127 . 2 A dom B → ran (B ↾ {A}) = ∅)
101, 9syl5eq 2081 1 A dom B → (B “ {A}) = ∅)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1242   ∈ wcel 1390   ∩ cin 2910  ∅c0 3218  {csn 3367  dom cdm 4288  ran crn 4289   ↾ cres 4290   “ cima 4291 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-eu 1900  df-mo 1901  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-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301 This theorem is referenced by:  fvun1  5182
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