Theorem ndmima 5105

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

Step	Hyp	Ref	Expression
1		df-ima 4732	. 2 ⊢ (𝐵 “ {𝐴}) = ran (𝐵 ↾ {𝐴})
2		dmres 5026	. . . . 5 ⊢ dom (𝐵 ↾ {𝐴}) = ({𝐴} ∩ dom 𝐵)
3		incom 3396	. . . . 5 ⊢ ({𝐴} ∩ dom 𝐵) = (dom 𝐵 ∩ {𝐴})
4	2, 3	eqtri 2250	. . . 4 ⊢ dom (𝐵 ↾ {𝐴}) = (dom 𝐵 ∩ {𝐴})
5		disjsn 3728	. . . . 5 ⊢ ((dom 𝐵 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ dom 𝐵)
6	5	biimpri 133	. . . 4 ⊢ (¬ 𝐴 ∈ dom 𝐵 → (dom 𝐵 ∩ {𝐴}) = ∅)
7	4, 6	eqtrid 2274	. . 3 ⊢ (¬ 𝐴 ∈ dom 𝐵 → dom (𝐵 ↾ {𝐴}) = ∅)
8		dm0rn0 4940	. . 3 ⊢ (dom (𝐵 ↾ {𝐴}) = ∅ ↔ ran (𝐵 ↾ {𝐴}) = ∅)
9	7, 8	sylib 122	. 2 ⊢ (¬ 𝐴 ∈ dom 𝐵 → ran (𝐵 ↾ {𝐴}) = ∅)
10	1, 9	eqtrid 2274	1 ⊢ (¬ 𝐴 ∈ dom 𝐵 → (𝐵 “ {𝐴}) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1395   ∈ wcel 2200   ∩ cin 3196  ∅c0 3491  {csn 3666  dom cdm 4719  ran crn 4720   ↾ cres 4721   “ cima 4722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-cnv 4727  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732

This theorem is referenced by:  fvun1  5702