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Theorem imadisj 4807
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 4465 . . 3 (𝐴𝐵) = ran (𝐴𝐵)
21eqeq1i 2096 . 2 ((𝐴𝐵) = ∅ ↔ ran (𝐴𝐵) = ∅)
3 dm0rn0 4666 . 2 (dom (𝐴𝐵) = ∅ ↔ ran (𝐴𝐵) = ∅)
4 dmres 4747 . . . 4 dom (𝐴𝐵) = (𝐵 ∩ dom 𝐴)
5 incom 3193 . . . 4 (𝐵 ∩ dom 𝐴) = (dom 𝐴𝐵)
64, 5eqtri 2109 . . 3 dom (𝐴𝐵) = (dom 𝐴𝐵)
76eqeq1i 2096 . 2 (dom (𝐴𝐵) = ∅ ↔ (dom 𝐴𝐵) = ∅)
82, 3, 73bitr2i 207 1 ((𝐴𝐵) = ∅ ↔ (dom 𝐴𝐵) = ∅)
Colors of variables: wff set class
Syntax hints:  wb 104   = wceq 1290  cin 2999  c0 3287  dom cdm 4452  ran crn 4453  cres 4454  cima 4455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-xp 4458  df-cnv 4460  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465
This theorem is referenced by:  fnimadisj  5147  fnimaeq0  5148  fimacnvdisj  5208
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