Theorem ndmima 3440

Description: The image of a singleton outside the domain is empty.

Assertion

Ref	Expression
ndmima	|- (-. A e. dom B -> (B"{A}) = (/))

Proof of Theorem ndmima

Step	Hyp	Ref	Expression
1		disjsn 2445	. . . . 5 |- ((dom B i^i {A}) = (/) <-> -. A e. dom B)
2	1	biimpr 152	. . . 4 |- (-. A e. dom B -> (dom B i^i {A}) = (/))
3		dmres 3386	. . . . 5 |- dom ( B |` {A}) = ({A} i^i dom B)
4		incom 2211	. . . . 5 |- ({A} i^i dom B) = (dom B i^i {A})
5	3, 4	eqtr 1498	. . . 4 |- dom ( B |` {A}) = (dom B i^i {A})
6	2, 5	syl5eq 1522	. . 3 |- (-. A e. dom B -> dom ( B |` {A}) = (/))
7		dm0rn0 3336	. . 3 |- (dom ( B |` {A}) = (/) <-> ran ( B |` {A}) = (/))
8	6, 7	sylib 198	. 2 |- (-. A e. dom B -> ran ( B |` {A}) = (/))
9		df-ima 3197	. 2 |- (B"{A}) = ran ( B |` {A})
10	8, 9	syl5eq 1522	1 |- (-. A e. dom B -> (B"{A}) = (/))

Syntax hints:  -. wn 2   -> wi 3   = wceq 958   e. wcel 960   i^i cin 2049  (/)c0 2283  {csn 2413  dom cdm 3176  ran crn 3177   |` cres 3178  "cima 3179

This theorem is referenced by:  funfv 3776

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-xp 3190  df-cnv 3192  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197

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