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Theorem dm0rn0 4641
Description: An empty domain implies an empty range. (Contributed by NM, 21-May-1998.)
Assertion
Ref Expression
dm0rn0 (dom 𝐴 = ∅ ↔ ran 𝐴 = ∅)

Proof of Theorem dm0rn0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 alnex 1433 . . . . . 6 (∀𝑥 ¬ ∃𝑦 𝑥𝐴𝑦 ↔ ¬ ∃𝑥𝑦 𝑥𝐴𝑦)
2 excom 1599 . . . . . 6 (∃𝑥𝑦 𝑥𝐴𝑦 ↔ ∃𝑦𝑥 𝑥𝐴𝑦)
31, 2xchbinx 642 . . . . 5 (∀𝑥 ¬ ∃𝑦 𝑥𝐴𝑦 ↔ ¬ ∃𝑦𝑥 𝑥𝐴𝑦)
4 alnex 1433 . . . . 5 (∀𝑦 ¬ ∃𝑥 𝑥𝐴𝑦 ↔ ¬ ∃𝑦𝑥 𝑥𝐴𝑦)
53, 4bitr4i 185 . . . 4 (∀𝑥 ¬ ∃𝑦 𝑥𝐴𝑦 ↔ ∀𝑦 ¬ ∃𝑥 𝑥𝐴𝑦)
6 noel 3288 . . . . . 6 ¬ 𝑥 ∈ ∅
76nbn 650 . . . . 5 (¬ ∃𝑦 𝑥𝐴𝑦 ↔ (∃𝑦 𝑥𝐴𝑦𝑥 ∈ ∅))
87albii 1404 . . . 4 (∀𝑥 ¬ ∃𝑦 𝑥𝐴𝑦 ↔ ∀𝑥(∃𝑦 𝑥𝐴𝑦𝑥 ∈ ∅))
9 noel 3288 . . . . . 6 ¬ 𝑦 ∈ ∅
109nbn 650 . . . . 5 (¬ ∃𝑥 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦𝑦 ∈ ∅))
1110albii 1404 . . . 4 (∀𝑦 ¬ ∃𝑥 𝑥𝐴𝑦 ↔ ∀𝑦(∃𝑥 𝑥𝐴𝑦𝑦 ∈ ∅))
125, 8, 113bitr3i 208 . . 3 (∀𝑥(∃𝑦 𝑥𝐴𝑦𝑥 ∈ ∅) ↔ ∀𝑦(∃𝑥 𝑥𝐴𝑦𝑦 ∈ ∅))
13 abeq1 2197 . . 3 ({𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = ∅ ↔ ∀𝑥(∃𝑦 𝑥𝐴𝑦𝑥 ∈ ∅))
14 abeq1 2197 . . 3 ({𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} = ∅ ↔ ∀𝑦(∃𝑥 𝑥𝐴𝑦𝑦 ∈ ∅))
1512, 13, 143bitr4i 210 . 2 ({𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = ∅ ↔ {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} = ∅)
16 df-dm 4438 . . 3 dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
1716eqeq1i 2095 . 2 (dom 𝐴 = ∅ ↔ {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = ∅)
18 dfrn2 4612 . . 3 ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
1918eqeq1i 2095 . 2 (ran 𝐴 = ∅ ↔ {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} = ∅)
2015, 17, 193bitr4i 210 1 (dom 𝐴 = ∅ ↔ ran 𝐴 = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 103  wal 1287   = wceq 1289  wex 1426  wcel 1438  {cab 2074  c0 3284   class class class wbr 3837  dom cdm 4428  ran crn 4429
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-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-cnv 4436  df-dm 4438  df-rn 4439
This theorem is referenced by:  rn0  4677  relrn0  4683  imadisj  4781  ndmima  4796  f00  5186  f0rn0  5189  2nd0  5898  map0b  6424
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