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Theorem imadisj 6099
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 ((𝐴𝐵) = ∅ ↔ (dom 𝐴𝐵) = ∅)

Proof of Theorem imadisj
StepHypRef Expression
1 df-ima 5701 . . 3 (𝐴𝐵) = ran (𝐴𝐵)
21eqeq1i 2739 . 2 ((𝐴𝐵) = ∅ ↔ ran (𝐴𝐵) = ∅)
3 dm0rn0 5937 . 2 (dom (𝐴𝐵) = ∅ ↔ ran (𝐴𝐵) = ∅)
4 dmres 6031 . . . 4 dom (𝐴𝐵) = (𝐵 ∩ dom 𝐴)
5 incom 4216 . . . 4 (𝐵 ∩ dom 𝐴) = (dom 𝐴𝐵)
64, 5eqtri 2762 . . 3 dom (𝐴𝐵) = (dom 𝐴𝐵)
76eqeq1i 2739 . 2 (dom (𝐴𝐵) = ∅ ↔ (dom 𝐴𝐵) = ∅)
82, 3, 73bitr2i 299 1 ((𝐴𝐵) = ∅ ↔ (dom 𝐴𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1536  cin 3961  c0 4338  dom cdm 5688  ran crn 5689  cres 5690  cima 5691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-xp 5694  df-cnv 5696  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701
This theorem is referenced by:  imadisjlnd  6100  ndmima  6123  fnimadisj  6700  fnimaeq0  6701  fimacnvdisj  6786  frxp2  8167  frxp3  8174  acndom2  10091  isf34lem5  10415  isf34lem7  10416  isf34lem6  10417  limsupgre  15513  isercolllem3  15699  pf1rcl  22368  cnconn  23445  1stcfb  23468  xkohaus  23676  qtopeu  23739  fbasrn  23907  mbflimsup  25714  preiman0  32724  eulerpartlemt  34352  erdszelem5  35179  fnwe2lem2  43039  imadisjld  44149
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