Theorem ima0 3426

Description: Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38.

Assertion

Ref	Expression
ima0	|- (A"(/)) = (/)

Proof of Theorem ima0

Step	Hyp	Ref	Expression
1		df-ima 3197	. 2 |- (A"(/)) = ran ( A |` (/))
2		res0 3377	. . 3 |- (A |` (/)) = (/)
3	2	rneqi 3346	. 2 |- ran ( A |` (/)) = ran (/)
4		rn0 3361	. 2 |- ran (/) = (/)
5	1, 3, 4	3eqtr 1502	1 |- (A"(/)) = (/)

Syntax hints:   = wceq 958  (/)c0 2283  ran crn 3177   |` cres 3178  "cima 3179

This theorem is referenced by:  relimasn 3431  fvprc 3727  mapudiscn 10498  eqindhome 10527

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-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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