Theorem dm0rn0 4996

Description: An empty domain implies an empty range. For a similar theorem for whether the domain and range are inhabited, see dmmrnm 4999. (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.

Step	Hyp	Ref	Expression
1		alnex 1552	. . . . . 6 ⊢ (∀𝑥 ¬ ∃𝑦 𝑥𝐴𝑦 ↔ ¬ ∃𝑥∃𝑦 𝑥𝐴𝑦)
2		excom 1716	. . . . . 6 ⊢ (∃𝑥∃𝑦 𝑥𝐴𝑦 ↔ ∃𝑦∃𝑥 𝑥𝐴𝑦)
3	1, 2	xchbinx 693	. . . . 5 ⊢ (∀𝑥 ¬ ∃𝑦 𝑥𝐴𝑦 ↔ ¬ ∃𝑦∃𝑥 𝑥𝐴𝑦)
4		alnex 1552	. . . . 5 ⊢ (∀𝑦 ¬ ∃𝑥 𝑥𝐴𝑦 ↔ ¬ ∃𝑦∃𝑥 𝑥𝐴𝑦)
5	3, 4	bitr4i 187	. . . 4 ⊢ (∀𝑥 ¬ ∃𝑦 𝑥𝐴𝑦 ↔ ∀𝑦 ¬ ∃𝑥 𝑥𝐴𝑦)
6		noel 3525	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
7	6	nbn 711	. . . . 5 ⊢ (¬ ∃𝑦 𝑥𝐴𝑦 ↔ (∃𝑦 𝑥𝐴𝑦 ↔ 𝑥 ∈ ∅))
8	7	albii 1523	. . . 4 ⊢ (∀𝑥 ¬ ∃𝑦 𝑥𝐴𝑦 ↔ ∀𝑥(∃𝑦 𝑥𝐴𝑦 ↔ 𝑥 ∈ ∅))
9		noel 3525	. . . . . 6 ⊢ ¬ 𝑦 ∈ ∅
10	9	nbn 711	. . . . 5 ⊢ (¬ ∃𝑥 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ↔ 𝑦 ∈ ∅))
11	10	albii 1523	. . . 4 ⊢ (∀𝑦 ¬ ∃𝑥 𝑥𝐴𝑦 ↔ ∀𝑦(∃𝑥 𝑥𝐴𝑦 ↔ 𝑦 ∈ ∅))
12	5, 8, 11	3bitr3i 210	. . 3 ⊢ (∀𝑥(∃𝑦 𝑥𝐴𝑦 ↔ 𝑥 ∈ ∅) ↔ ∀𝑦(∃𝑥 𝑥𝐴𝑦 ↔ 𝑦 ∈ ∅))
13		abeq1 2348	. . 3 ⊢ ({𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = ∅ ↔ ∀𝑥(∃𝑦 𝑥𝐴𝑦 ↔ 𝑥 ∈ ∅))
14		abeq1 2348	. . 3 ⊢ ({𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} = ∅ ↔ ∀𝑦(∃𝑥 𝑥𝐴𝑦 ↔ 𝑦 ∈ ∅))
15	12, 13, 14	3bitr4i 212	. 2 ⊢ ({𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = ∅ ↔ {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} = ∅)
16		df-dm 4782	. . 3 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
17	16	eqeq1i 2246	. 2 ⊢ (dom 𝐴 = ∅ ↔ {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = ∅)
18		dfrn2 4966	. . 3 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
19	18	eqeq1i 2246	. 2 ⊢ (ran 𝐴 = ∅ ↔ {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} = ∅)
20	15, 17, 19	3bitr4i 212	1 ⊢ (dom 𝐴 = ∅ ↔ ran 𝐴 = ∅)

Syntax hints:  ¬ wn 3   ↔ wb 105  ∀wal 1400   = wceq 1402  ∃wex 1545   ∈ wcel 2209  {cab 2224  ∅c0 3520   class class class wbr 4128  dom cdm 4772  ran crn 4773

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129  df-opab 4191  df-cnv 4780  df-dm 4782  df-rn 4783

This theorem is referenced by:  rn0  5036  relrn0  5042  imadisj  5147  ndmima  5162  f00  5582  f0rn0  5585  2nd0  6373  map0b  6962