Theorem imadisj 4901

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 4552	. . 3 |- ( A " B ) = ran ( A |` B )
2	1	eqeq1i 2147	. 2 |- ( ( A " B ) = (/) <-> ran ( A |` B ) = (/) )
3		dm0rn0 4756	. 2 |- ( dom ( A |` B ) = (/) <-> ran ( A |` B ) = (/) )
4		dmres 4840	. . . 4 |- dom ( A |` B ) = ( B i^i dom A )
5		incom 3268	. . . 4 |- ( B i^i dom A ) = ( dom A i^i B )
6	4, 5	eqtri 2160	. . 3 |- dom ( A |` B ) = ( dom A i^i B )
7	6	eqeq1i 2147	. 2 |- ( dom ( A |` B ) = (/) <-> ( dom A i^i B ) = (/) )
8	2, 3, 7	3bitr2i 207	1 |- ( ( A " B ) = (/) <-> ( dom A i^i B ) = (/) )

Syntax hints:   <-> wb 104   = wceq 1331   i^i cin 3070   (/)c0 3363   dom cdm 4539   ran crn 4540   |` cres 4541   "cima 4542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552

This theorem is referenced by:  fnimadisj  5243  fnimaeq0  5244  fimacnvdisj  5307