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Theorem imadisj 5519
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 5156 . . 3 (𝐴𝐵) = ran (𝐴𝐵)
21eqeq1i 2656 . 2 ((𝐴𝐵) = ∅ ↔ ran (𝐴𝐵) = ∅)
3 dm0rn0 5374 . 2 (dom (𝐴𝐵) = ∅ ↔ ran (𝐴𝐵) = ∅)
4 dmres 5454 . . . 4 dom (𝐴𝐵) = (𝐵 ∩ dom 𝐴)
5 incom 3838 . . . 4 (𝐵 ∩ dom 𝐴) = (dom 𝐴𝐵)
64, 5eqtri 2673 . . 3 dom (𝐴𝐵) = (dom 𝐴𝐵)
76eqeq1i 2656 . 2 (dom (𝐴𝐵) = ∅ ↔ (dom 𝐴𝐵) = ∅)
82, 3, 73bitr2i 288 1 ((𝐴𝐵) = ∅ ↔ (dom 𝐴𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1523  cin 3606  c0 3948  dom cdm 5143  ran crn 5144  cres 5145  cima 5146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156
This theorem is referenced by:  ndmima  5537  fnimadisj  6050  fnimaeq0  6051  fimacnvdisj  6121  acndom2  8915  isf34lem5  9238  isf34lem7  9239  isf34lem6  9240  limsupgre  14256  isercolllem3  14441  pf1rcl  19761  cnconn  21273  1stcfb  21296  xkohaus  21504  qtopeu  21567  fbasrn  21735  mbflimsup  23478  eulerpartlemt  30561  erdszelem5  31303  fnwe2lem2  37938  imadisjld  38775  imadisjlnd  38776  wnefimgd  38777
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