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Theorem imadisj 5215
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
StepHypRef Expression
1 df-ima 4883 . . 3  |-  ( A
" B )  =  ran  ( A  |`  B )
21eqeq1i 2442 . 2  |-  ( ( A " B )  =  (/)  <->  ran  ( A  |`  B )  =  (/) )
3 dm0rn0 5078 . 2  |-  ( dom  ( A  |`  B )  =  (/)  <->  ran  ( A  |`  B )  =  (/) )
4 dmres 5159 . . . 4  |-  dom  ( A  |`  B )  =  ( B  i^i  dom  A )
5 incom 3525 . . . 4  |-  ( B  i^i  dom  A )  =  ( dom  A  i^i  B )
64, 5eqtri 2455 . . 3  |-  dom  ( A  |`  B )  =  ( dom  A  i^i  B )
76eqeq1i 2442 . 2  |-  ( dom  ( A  |`  B )  =  (/)  <->  ( dom  A  i^i  B )  =  (/) )
82, 3, 73bitr2i 265 1  |-  ( ( A " B )  =  (/)  <->  ( dom  A  i^i  B )  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1652    i^i cin 3311   (/)c0 3620   dom cdm 4870   ran crn 4871    |` cres 4872   "cima 4873
This theorem is referenced by:  fnimadisj  5557  fnimaeq0  5558  fimacnvdisj  5613  acndom2  7927  isf34lem5  8250  isf34lem7  8251  isf34lem6  8252  limsupgre  12267  isercolllem3  12452  cnconn  17477  1stcfb  17500  xkohaus  17677  qtopeu  17740  fbasrn  17908  mbflimsup  19550  pf1rcl  19961  erdszelem5  24873  fnwe2lem2  27107
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883
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