Theorem imadisj 5028

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 4673	. . 3 |- ( A " B ) = ran ( A |` B )
2	1	eqeq1i 2201	. 2 |- ( ( A " B ) = (/) <-> ran ( A |` B ) = (/) )
3		dm0rn0 4880	. 2 |- ( dom ( A |` B ) = (/) <-> ran ( A |` B ) = (/) )
4		dmres 4964	. . . 4 |- dom ( A |` B ) = ( B i^i dom A )
5		incom 3352	. . . 4 |- ( B i^i dom A ) = ( dom A i^i B )
6	4, 5	eqtri 2214	. . 3 |- dom ( A |` B ) = ( dom A i^i B )
7	6	eqeq1i 2201	. 2 |- ( dom ( A |` B ) = (/) <-> ( dom A i^i B ) = (/) )
8	2, 3, 7	3bitr2i 208	1 |- ( ( A " B ) = (/) <-> ( dom A i^i B ) = (/) )

Syntax hints:   <-> wb 105   = wceq 1364   i^i cin 3153   (/)c0 3447   dom cdm 4660   ran crn 4661   |` cres 4662   "cima 4663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-xp 4666  df-cnv 4668  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673

This theorem is referenced by:  fnimadisj  5375  fnimaeq0  5376  fimacnvdisj  5439