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Theorem imadisj 4781
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 4441 . . 3 (𝐴𝐵) = ran (𝐴𝐵)
21eqeq1i 2095 . 2 ((𝐴𝐵) = ∅ ↔ ran (𝐴𝐵) = ∅)
3 dm0rn0 4641 . 2 (dom (𝐴𝐵) = ∅ ↔ ran (𝐴𝐵) = ∅)
4 dmres 4721 . . . 4 dom (𝐴𝐵) = (𝐵 ∩ dom 𝐴)
5 incom 3190 . . . 4 (𝐵 ∩ dom 𝐴) = (dom 𝐴𝐵)
64, 5eqtri 2108 . . 3 dom (𝐴𝐵) = (dom 𝐴𝐵)
76eqeq1i 2095 . 2 (dom (𝐴𝐵) = ∅ ↔ (dom 𝐴𝐵) = ∅)
82, 3, 73bitr2i 206 1 ((𝐴𝐵) = ∅ ↔ (dom 𝐴𝐵) = ∅)
Colors of variables: wff set class
Syntax hints:  wb 103   = wceq 1289  cin 2996  c0 3284  dom cdm 4428  ran crn 4429  cres 4430  cima 4431
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-cnv 4436  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441
This theorem is referenced by:  fnimadisj  5120  fnimaeq0  5121  fimacnvdisj  5179
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