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Theorem imadisj 5005
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 4654 . . 3 (𝐴𝐵) = ran (𝐴𝐵)
21eqeq1i 2197 . 2 ((𝐴𝐵) = ∅ ↔ ran (𝐴𝐵) = ∅)
3 dm0rn0 4859 . 2 (dom (𝐴𝐵) = ∅ ↔ ran (𝐴𝐵) = ∅)
4 dmres 4943 . . . 4 dom (𝐴𝐵) = (𝐵 ∩ dom 𝐴)
5 incom 3342 . . . 4 (𝐵 ∩ dom 𝐴) = (dom 𝐴𝐵)
64, 5eqtri 2210 . . 3 dom (𝐴𝐵) = (dom 𝐴𝐵)
76eqeq1i 2197 . 2 (dom (𝐴𝐵) = ∅ ↔ (dom 𝐴𝐵) = ∅)
82, 3, 73bitr2i 208 1 ((𝐴𝐵) = ∅ ↔ (dom 𝐴𝐵) = ∅)
Colors of variables: wff set class
Syntax hints:  wb 105   = wceq 1364  cin 3143  c0 3437  dom cdm 4641  ran crn 4642  cres 4643  cima 4644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4647  df-cnv 4649  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654
This theorem is referenced by:  fnimadisj  5351  fnimaeq0  5352  fimacnvdisj  5415
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