Theorem ndmima 3434

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 2441	. . . . 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 3380	. . . . 5 |- dom ( B |` {A}) = ({A} i^i dom B)
4		incom 2208	. . . . 5 |- ({A} i^i dom B) = (dom B i^i {A})
5	3, 4	eqtr 1495	. . . 4 |- dom ( B |` {A}) = (dom B i^i {A})
6	2, 5	syl5eq 1519	. . 3 |- (-. A e. dom B -> dom ( B |` {A}) = (/))
7		dm0rn0 3330	. . 3 |- (dom ( B |` {A}) = (/) <-> ran ( B |` {A}) = (/))
8	6, 7	sylib 198	. 2 |- (-. A e. dom B -> ran ( B |` {A}) = (/))
9		df-ima 3191	. 2 |- (B"{A}) = ran ( B |` {A})
10	8, 9	syl5eq 1519	1 |- (-. A e. dom B -> (B"{A}) = (/))

Syntax hints:  -. wn 2   -> wi 3   = wceq 956   e. wcel 958   i^i cin 2046  (/)c0 2280  {csn 2409  dom cdm 3170  ran crn 3171   |` cres 3172  "cima 3173

This theorem is referenced by:  funfv 3770

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191

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