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Theorem dm0rn0 4574
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 1429 . . . . . 6 (∀𝑥 ¬ ∃𝑦 𝑥𝐴𝑦 ↔ ¬ ∃𝑥𝑦 𝑥𝐴𝑦)
2 excom 1595 . . . . . 6 (∃𝑥𝑦 𝑥𝐴𝑦 ↔ ∃𝑦𝑥 𝑥𝐴𝑦)
31, 2xchbinx 640 . . . . 5 (∀𝑥 ¬ ∃𝑦 𝑥𝐴𝑦 ↔ ¬ ∃𝑦𝑥 𝑥𝐴𝑦)
4 alnex 1429 . . . . 5 (∀𝑦 ¬ ∃𝑥 𝑥𝐴𝑦 ↔ ¬ ∃𝑦𝑥 𝑥𝐴𝑦)
53, 4bitr4i 185 . . . 4 (∀𝑥 ¬ ∃𝑦 𝑥𝐴𝑦 ↔ ∀𝑦 ¬ ∃𝑥 𝑥𝐴𝑦)
6 noel 3256 . . . . . 6 ¬ 𝑥 ∈ ∅
76nbn 648 . . . . 5 (¬ ∃𝑦 𝑥𝐴𝑦 ↔ (∃𝑦 𝑥𝐴𝑦𝑥 ∈ ∅))
87albii 1400 . . . 4 (∀𝑥 ¬ ∃𝑦 𝑥𝐴𝑦 ↔ ∀𝑥(∃𝑦 𝑥𝐴𝑦𝑥 ∈ ∅))
9 noel 3256 . . . . . 6 ¬ 𝑦 ∈ ∅
109nbn 648 . . . . 5 (¬ ∃𝑥 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦𝑦 ∈ ∅))
1110albii 1400 . . . 4 (∀𝑦 ¬ ∃𝑥 𝑥𝐴𝑦 ↔ ∀𝑦(∃𝑥 𝑥𝐴𝑦𝑦 ∈ ∅))
125, 8, 113bitr3i 208 . . 3 (∀𝑥(∃𝑦 𝑥𝐴𝑦𝑥 ∈ ∅) ↔ ∀𝑦(∃𝑥 𝑥𝐴𝑦𝑦 ∈ ∅))
13 abeq1 2189 . . 3 ({𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = ∅ ↔ ∀𝑥(∃𝑦 𝑥𝐴𝑦𝑥 ∈ ∅))
14 abeq1 2189 . . 3 ({𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} = ∅ ↔ ∀𝑦(∃𝑥 𝑥𝐴𝑦𝑦 ∈ ∅))
1512, 13, 143bitr4i 210 . 2 ({𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = ∅ ↔ {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} = ∅)
16 df-dm 4375 . . 3 dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
1716eqeq1i 2089 . 2 (dom 𝐴 = ∅ ↔ {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = ∅)
18 dfrn2 4545 . . 3 ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
1918eqeq1i 2089 . 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 1283   = wceq 1285  wex 1422  wcel 1434  {cab 2068  c0 3252   class class class wbr 3787  dom cdm 4365  ran crn 4366
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788  df-opab 3842  df-cnv 4373  df-dm 4375  df-rn 4376
This theorem is referenced by:  rn0  4610  relrn0  4616  imadisj  4711  ndmima  4726  f00  5106  2nd0  5797
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