Theorem imadisj 3414

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 3186	. . 3 |- (A"B) = ran ( A |` B)
2	1	eqeq1i 1479	. 2 |- ((A"B) = (/) <-> ran ( A |` B) = (/))
3		dm0rn0 3325	. 2 |- (dom ( A |` B) = (/) <-> ran ( A |` B) = (/))
4		dmres 3372	. . . 4 |- dom ( A |` B) = (B i^i dom A)
5		incom 2204	. . . 4 |- (B i^i dom A) = (dom A i^i B)
6	4, 5	eqtr 1492	. . 3 |- dom ( A |` B) = (dom A i^i B)
7	6	eqeq1i 1479	. 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 954   i^i cin 2042  (/)c0 2276  dom cdm 3165  ran crn 3166   |` cres 3167  "cima 3168

This theorem is referenced by:  funimadisj 3598  fimacnvdisj 3640

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-xp 3179  df-cnv 3181  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186

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