Theorem imadisj 5129

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 4767	. . 3 |- ( A " B ) = ran ( A |` B )
2	1	eqeq1i 2242	. 2 |- ( ( A " B ) = (/) <-> ran ( A |` B ) = (/) )
3		dm0rn0 4978	. 2 |- ( dom ( A |` B ) = (/) <-> ran ( A |` B ) = (/) )
4		dmres 5064	. . . 4 |- dom ( A |` B ) = ( B i^i dom A )
5		incom 3415	. . . 4 |- ( B i^i dom A ) = ( dom A i^i B )
6	4, 5	eqtri 2255	. . 3 |- dom ( A |` B ) = ( dom A i^i B )
7	6	eqeq1i 2242	. 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 1398   i^i cin 3213   (/)c0 3512   dom cdm 4754   ran crn 4755   |` cres 4756   "cima 4757

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-xp 4760  df-cnv 4762  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767

This theorem is referenced by:  fnimadisj  5484  fnimaeq0  5485  fimacnvdisj  5556