Theorem dm0rn0OLD 5906

Description: Obsolete version of dm0rn0 5905 as of 24-Jan-2026. (Contributed by NM, 21-May-1998.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
dm0rn0OLD	⊢ (dom 𝐴 = ∅ ↔ ran 𝐴 = ∅)

Proof of Theorem dm0rn0OLD

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		alnex 1813	. . . . . 6 ⊢ (∀𝑥 ¬ ∃𝑦 𝑥𝐴𝑦 ↔ ¬ ∃𝑥∃𝑦 𝑥𝐴𝑦)
2		excom 2198	. . . . . 6 ⊢ (∃𝑥∃𝑦 𝑥𝐴𝑦 ↔ ∃𝑦∃𝑥 𝑥𝐴𝑦)
3	1, 2	xchbinx 337	. . . . 5 ⊢ (∀𝑥 ¬ ∃𝑦 𝑥𝐴𝑦 ↔ ¬ ∃𝑦∃𝑥 𝑥𝐴𝑦)
4		alnex 1813	. . . . 5 ⊢ (∀𝑦 ¬ ∃𝑥 𝑥𝐴𝑦 ↔ ¬ ∃𝑦∃𝑥 𝑥𝐴𝑦)
5	3, 4	bitr4i 281	. . . 4 ⊢ (∀𝑥 ¬ ∃𝑦 𝑥𝐴𝑦 ↔ ∀𝑦 ¬ ∃𝑥 𝑥𝐴𝑦)
6		noel 4283	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
7	6	nbn 375	. . . . 5 ⊢ (¬ ∃𝑦 𝑥𝐴𝑦 ↔ (∃𝑦 𝑥𝐴𝑦 ↔ 𝑥 ∈ ∅))
8	7	albii 1851	. . . 4 ⊢ (∀𝑥 ¬ ∃𝑦 𝑥𝐴𝑦 ↔ ∀𝑥(∃𝑦 𝑥𝐴𝑦 ↔ 𝑥 ∈ ∅))
9		noel 4283	. . . . . 6 ⊢ ¬ 𝑦 ∈ ∅
10	9	nbn 375	. . . . 5 ⊢ (¬ ∃𝑥 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ↔ 𝑦 ∈ ∅))
11	10	albii 1851	. . . 4 ⊢ (∀𝑦 ¬ ∃𝑥 𝑥𝐴𝑦 ↔ ∀𝑦(∃𝑥 𝑥𝐴𝑦 ↔ 𝑦 ∈ ∅))
12	5, 8, 11	3bitr3i 304	. . 3 ⊢ (∀𝑥(∃𝑦 𝑥𝐴𝑦 ↔ 𝑥 ∈ ∅) ↔ ∀𝑦(∃𝑥 𝑥𝐴𝑦 ↔ 𝑦 ∈ ∅))
13		eqabcb 2900	. . 3 ⊢ ({𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = ∅ ↔ ∀𝑥(∃𝑦 𝑥𝐴𝑦 ↔ 𝑥 ∈ ∅))
14		eqabcb 2900	. . 3 ⊢ ({𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} = ∅ ↔ ∀𝑦(∃𝑥 𝑥𝐴𝑦 ↔ 𝑦 ∈ ∅))
15	12, 13, 14	3bitr4i 306	. 2 ⊢ ({𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = ∅ ↔ {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} = ∅)
16		df-dm 5660	. . 3 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
17	16	eqeq1i 2765	. 2 ⊢ (dom 𝐴 = ∅ ↔ {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = ∅)
18		dfrn2 5869	. . 3 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
19	18	eqeq1i 2765	. 2 ⊢ (ran 𝐴 = ∅ ↔ {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} = ∅)
20	15, 17, 19	3bitr4i 306	1 ⊢ (dom 𝐴 = ∅ ↔ ran 𝐴 = ∅)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209  ∀wal 1567   = wceq 1569  ∃wex 1811   ∈ wcel 2144  {cab 2738  ∅c0 4278   class class class wbr 5102  dom cdm 5650  ran crn 5651

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-cnv 5658  df-dm 5660  df-rn 5661

This theorem is used by: (None)