Theorem imadisj 5031

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 4676	. . 3 |- ( A " B ) = ran ( A |` B )
2	1	eqeq1i 2204	. 2 |- ( ( A " B ) = (/) <-> ran ( A |` B ) = (/) )
3		dm0rn0 4883	. 2 |- ( dom ( A |` B ) = (/) <-> ran ( A |` B ) = (/) )
4		dmres 4967	. . . 4 |- dom ( A |` B ) = ( B i^i dom A )
5		incom 3355	. . . 4 |- ( B i^i dom A ) = ( dom A i^i B )
6	4, 5	eqtri 2217	. . 3 |- dom ( A |` B ) = ( dom A i^i B )
7	6	eqeq1i 2204	. 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 3156   (/)c0 3450   dom cdm 4663   ran crn 4664   |` cres 4665   "cima 4666

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-xp 4669  df-cnv 4671  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676

This theorem is referenced by:  fnimadisj  5378  fnimaeq0  5379  fimacnvdisj  5442