Theorem imadisj 3422

Description: A class whose image under another is empty is disjoint with the other's domain. (Contributed by FL, 24-Jan-2007.)

Assertion

Ref	Expression
imadisj	|- ((A"B) = (/) <-> (dom A i^i B) = (/))

Proof of Theorem imadisj

Step	Hyp	Ref	Expression
1		df-ima 3191	. . 3 |- (A"B) = ran ( A |` B)
2	1	eqeq1i 1482	. 2 |- ((A"B) = (/) <-> ran ( A |` B) = (/))
3		dm0rn0 3330	. 2 |- (dom ( A |` B) = (/) <-> ran ( A |` B) = (/))
4		dmres 3380	. . . 4 |- dom ( A |` B) = (B i^i dom A)
5		incom 2208	. . . 4 |- (B i^i dom A) = (dom A i^i B)
6	4, 5	eqtr 1495	. . 3 |- dom ( A |` B) = (dom A i^i B)
7	6	eqeq1i 1482	. 2 |- (dom ( A |` B) = (/) <-> (dom A i^i B) = (/))
8	2, 3, 7	3bitr2 179	1 |- ((A"B) = (/) <-> (dom A i^i B) = (/))

Syntax hints:   <-> wb 146   = wceq 956   i^i cin 2046  (/)c0 2280  dom cdm 3170  ran crn 3171   |` cres 3172  "cima 3173

This theorem is referenced by:  funimadisj 3606  fimacnvdisj 3649

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-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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