Theorem imadisj 5484

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 5127	. . 3 |- ( A " B ) = ran ( A |` B )
2	1	eqeq1i 2627	. 2 |- ( ( A " B ) = (/) <-> ran ( A |` B ) = (/) )
3		dm0rn0 5342	. 2 |- ( dom ( A |` B ) = (/) <-> ran ( A |` B ) = (/) )
4		dmres 5419	. . . 4 |- dom ( A |` B ) = ( B i^i dom A )
5		incom 3805	. . . 4 |- ( B i^i dom A ) = ( dom A i^i B )
6	4, 5	eqtri 2644	. . 3 |- dom ( A |` B ) = ( dom A i^i B )
7	6	eqeq1i 2627	. 2 |- ( dom ( A |` B ) = (/) <-> ( dom A i^i B ) = (/) )
8	2, 3, 7	3bitr2i 288	1 |- ( ( A " B ) = (/) <-> ( dom A i^i B ) = (/) )

Syntax hints:   <-> wb 196   = wceq 1483   i^i cin 3573   (/)c0 3915   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  ndmima  5502  fnimadisj  6012  fnimaeq0  6013  fimacnvdisj  6083  acndom2  8877  isf34lem5  9200  isf34lem7  9201  isf34lem6  9202  limsupgre  14212  isercolllem3  14397  pf1rcl  19713  cnconn  21225  1stcfb  21248  xkohaus  21456  qtopeu  21519  fbasrn  21688  mbflimsup  23433  eulerpartlemt  30433  erdszelem5  31177  fnwe2lem2  37621  imadisjld  38458  imadisjlnd  38459  wnefimgd  38460