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Theorem dm0rn0 4724
Description: An empty domain implies an empty range. For a similar theorem for whether the domain and range are inhabited, see dmmrnm 4726. (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 1458 . . . . . 6 (∀𝑥 ¬ ∃𝑦 𝑥𝐴𝑦 ↔ ¬ ∃𝑥𝑦 𝑥𝐴𝑦)
2 excom 1625 . . . . . 6 (∃𝑥𝑦 𝑥𝐴𝑦 ↔ ∃𝑦𝑥 𝑥𝐴𝑦)
31, 2xchbinx 654 . . . . 5 (∀𝑥 ¬ ∃𝑦 𝑥𝐴𝑦 ↔ ¬ ∃𝑦𝑥 𝑥𝐴𝑦)
4 alnex 1458 . . . . 5 (∀𝑦 ¬ ∃𝑥 𝑥𝐴𝑦 ↔ ¬ ∃𝑦𝑥 𝑥𝐴𝑦)
53, 4bitr4i 186 . . . 4 (∀𝑥 ¬ ∃𝑦 𝑥𝐴𝑦 ↔ ∀𝑦 ¬ ∃𝑥 𝑥𝐴𝑦)
6 noel 3335 . . . . . 6 ¬ 𝑥 ∈ ∅
76nbn 671 . . . . 5 (¬ ∃𝑦 𝑥𝐴𝑦 ↔ (∃𝑦 𝑥𝐴𝑦𝑥 ∈ ∅))
87albii 1429 . . . 4 (∀𝑥 ¬ ∃𝑦 𝑥𝐴𝑦 ↔ ∀𝑥(∃𝑦 𝑥𝐴𝑦𝑥 ∈ ∅))
9 noel 3335 . . . . . 6 ¬ 𝑦 ∈ ∅
109nbn 671 . . . . 5 (¬ ∃𝑥 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦𝑦 ∈ ∅))
1110albii 1429 . . . 4 (∀𝑦 ¬ ∃𝑥 𝑥𝐴𝑦 ↔ ∀𝑦(∃𝑥 𝑥𝐴𝑦𝑦 ∈ ∅))
125, 8, 113bitr3i 209 . . 3 (∀𝑥(∃𝑦 𝑥𝐴𝑦𝑥 ∈ ∅) ↔ ∀𝑦(∃𝑥 𝑥𝐴𝑦𝑦 ∈ ∅))
13 abeq1 2225 . . 3 ({𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = ∅ ↔ ∀𝑥(∃𝑦 𝑥𝐴𝑦𝑥 ∈ ∅))
14 abeq1 2225 . . 3 ({𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} = ∅ ↔ ∀𝑦(∃𝑥 𝑥𝐴𝑦𝑦 ∈ ∅))
1512, 13, 143bitr4i 211 . 2 ({𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = ∅ ↔ {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} = ∅)
16 df-dm 4517 . . 3 dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
1716eqeq1i 2123 . 2 (dom 𝐴 = ∅ ↔ {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = ∅)
18 dfrn2 4695 . . 3 ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
1918eqeq1i 2123 . 2 (ran 𝐴 = ∅ ↔ {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} = ∅)
2015, 17, 193bitr4i 211 1 (dom 𝐴 = ∅ ↔ ran 𝐴 = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 104  wal 1312   = wceq 1314  wex 1451  wcel 1463  {cab 2101  c0 3331   class class class wbr 3897  dom cdm 4507  ran crn 4508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-cnv 4515  df-dm 4517  df-rn 4518
This theorem is referenced by:  rn0  4763  relrn0  4769  imadisj  4869  ndmima  4884  f00  5282  f0rn0  5285  2nd0  6009  map0b  6547
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