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Theorem imadisj 5443
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 5087 . . 3 (𝐴𝐵) = ran (𝐴𝐵)
21eqeq1i 2626 . 2 ((𝐴𝐵) = ∅ ↔ ran (𝐴𝐵) = ∅)
3 dm0rn0 5302 . 2 (dom (𝐴𝐵) = ∅ ↔ ran (𝐴𝐵) = ∅)
4 dmres 5378 . . . 4 dom (𝐴𝐵) = (𝐵 ∩ dom 𝐴)
5 incom 3783 . . . 4 (𝐵 ∩ dom 𝐴) = (dom 𝐴𝐵)
64, 5eqtri 2643 . . 3 dom (𝐴𝐵) = (dom 𝐴𝐵)
76eqeq1i 2626 . 2 (dom (𝐴𝐵) = ∅ ↔ (dom 𝐴𝐵) = ∅)
82, 3, 73bitr2i 288 1 ((𝐴𝐵) = ∅ ↔ (dom 𝐴𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  cin 3554  c0 3891  dom cdm 5074  ran crn 5075  cres 5076  cima 5077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-cnv 5082  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087
This theorem is referenced by:  ndmima  5461  fnimadisj  5969  fnimaeq0  5970  fimacnvdisj  6040  acndom2  8821  isf34lem5  9144  isf34lem7  9145  isf34lem6  9146  limsupgre  14146  isercolllem3  14331  pf1rcl  19632  cnconn  21135  1stcfb  21158  xkohaus  21366  qtopeu  21429  fbasrn  21598  mbflimsup  23339  eulerpartlemt  30214  erdszelem5  30885  fnwe2lem2  37101  imadisjld  37940  imadisjlnd  37941  wnefimgd  37942
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