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Theorem dm0rn0 4475
 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 1365 . . . . . 6 (x ¬ y xAy ↔ ¬ xy xAy)
2 excom 1532 . . . . . 6 (xy xAyyx xAy)
31, 2xchbinx 594 . . . . 5 (x ¬ y xAy ↔ ¬ yx xAy)
4 alnex 1365 . . . . 5 (y ¬ x xAy ↔ ¬ yx xAy)
53, 4bitr4i 176 . . . 4 (x ¬ y xAyy ¬ x xAy)
6 noel 3201 . . . . . 6 ¬ x
76nbn 602 . . . . 5 y xAy ↔ (y xAyx ∅))
87albii 1335 . . . 4 (x ¬ y xAyx(y xAyx ∅))
9 noel 3201 . . . . . 6 ¬ y
109nbn 602 . . . . 5 x xAy ↔ (x xAyy ∅))
1110albii 1335 . . . 4 (y ¬ x xAyy(x xAyy ∅))
125, 8, 113bitr3i 199 . . 3 (x(y xAyx ∅) ↔ y(x xAyy ∅))
13 abeq1 2125 . . 3 ({xy xAy} = ∅ ↔ x(y xAyx ∅))
14 abeq1 2125 . . 3 ({yx xAy} = ∅ ↔ y(x xAyy ∅))
1512, 13, 143bitr4i 201 . 2 ({xy xAy} = ∅ ↔ {yx xAy} = ∅)
16 df-dm 4278 . . 3 dom A = {xy xAy}
1716eqeq1i 2025 . 2 (dom A = ∅ ↔ {xy xAy} = ∅)
18 dfrn2 4446 . . 3 ran A = {yx xAy}
1918eqeq1i 2025 . 2 (ran A = ∅ ↔ {yx xAy} = ∅)
2015, 17, 193bitr4i 201 1 (dom A = ∅ ↔ ran A = ∅)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ↔ wb 98  ∀wal 1224   = wceq 1226  ∃wex 1358   ∈ wcel 1370  {cab 2004  ∅c0 3197   class class class wbr 3734  dom cdm 4268  ran crn 4269 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914 This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-nul 3198  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-cnv 4276  df-dm 4278  df-rn 4279 This theorem is referenced by:  rn0  4511  relrn0  4517  imadisj  4610  ndmima  4625  f00  5002  2nd0  5691
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