Theorem reldmm 4941

Description: A relation is inhabited iff its domain is inhabited. (Contributed by Jim Kingdon, 30-Jan-2026.)

Assertion

Ref	Expression
reldmm	⊢ (Rel 𝐴 → (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑦 𝑦 ∈ dom 𝐴))

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴

Proof of Theorem reldmm

Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1w 2290	. . 3 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
2	1	cbvexv 1965	. 2 ⊢ (∃𝑤 𝑤 ∈ 𝐴 ↔ ∃𝑥 𝑥 ∈ 𝐴)
3		elrel 4820	. . . . . . . 8 ⊢ ((Rel 𝐴 ∧ 𝑤 ∈ 𝐴) → ∃𝑦∃𝑧 𝑤 = ⟨𝑦, 𝑧⟩)
4		eleq1 2292	. . . . . . . . . . . 12 ⊢ (𝑤 = ⟨𝑦, 𝑧⟩ → (𝑤 ∈ 𝐴 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
5	4	biimpd 144	. . . . . . . . . . 11 ⊢ (𝑤 = ⟨𝑦, 𝑧⟩ → (𝑤 ∈ 𝐴 → ⟨𝑦, 𝑧⟩ ∈ 𝐴))
6	5	eximi 1646	. . . . . . . . . 10 ⊢ (∃𝑧 𝑤 = ⟨𝑦, 𝑧⟩ → ∃𝑧(𝑤 ∈ 𝐴 → ⟨𝑦, 𝑧⟩ ∈ 𝐴))
7		nfv 1574	. . . . . . . . . . 11 ⊢ Ⅎ𝑧 𝑤 ∈ 𝐴
8	7	19.37-1 1720	. . . . . . . . . 10 ⊢ (∃𝑧(𝑤 ∈ 𝐴 → ⟨𝑦, 𝑧⟩ ∈ 𝐴) → (𝑤 ∈ 𝐴 → ∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝐴))
9	6, 8	syl 14	. . . . . . . . 9 ⊢ (∃𝑧 𝑤 = ⟨𝑦, 𝑧⟩ → (𝑤 ∈ 𝐴 → ∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝐴))
10	9	eximi 1646	. . . . . . . 8 ⊢ (∃𝑦∃𝑧 𝑤 = ⟨𝑦, 𝑧⟩ → ∃𝑦(𝑤 ∈ 𝐴 → ∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝐴))
11		nfv 1574	. . . . . . . . 9 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐴
12	11	19.37-1 1720	. . . . . . . 8 ⊢ (∃𝑦(𝑤 ∈ 𝐴 → ∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝐴) → (𝑤 ∈ 𝐴 → ∃𝑦∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝐴))
13	3, 10, 12	3syl 17	. . . . . . 7 ⊢ ((Rel 𝐴 ∧ 𝑤 ∈ 𝐴) → (𝑤 ∈ 𝐴 → ∃𝑦∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝐴))
14	13	syldbl2 1326	. . . . . 6 ⊢ ((Rel 𝐴 ∧ 𝑤 ∈ 𝐴) → ∃𝑦∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝐴)
15		vex 2802	. . . . . . . 8 ⊢ 𝑦 ∈ V
16	15	eldm2 4920	. . . . . . 7 ⊢ (𝑦 ∈ dom 𝐴 ↔ ∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝐴)
17	16	exbii 1651	. . . . . 6 ⊢ (∃𝑦 𝑦 ∈ dom 𝐴 ↔ ∃𝑦∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝐴)
18	14, 17	sylibr 134	. . . . 5 ⊢ ((Rel 𝐴 ∧ 𝑤 ∈ 𝐴) → ∃𝑦 𝑦 ∈ dom 𝐴)
19	18	ex 115	. . . 4 ⊢ (Rel 𝐴 → (𝑤 ∈ 𝐴 → ∃𝑦 𝑦 ∈ dom 𝐴))
20	19	exlimdv 1865	. . 3 ⊢ (Rel 𝐴 → (∃𝑤 𝑤 ∈ 𝐴 → ∃𝑦 𝑦 ∈ dom 𝐴))
21		elex2 2816	. . . . 5 ⊢ (⟨𝑦, 𝑧⟩ ∈ 𝐴 → ∃𝑤 𝑤 ∈ 𝐴)
22	21	exlimivv 1943	. . . 4 ⊢ (∃𝑦∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝐴 → ∃𝑤 𝑤 ∈ 𝐴)
23	17, 22	sylbi 121	. . 3 ⊢ (∃𝑦 𝑦 ∈ dom 𝐴 → ∃𝑤 𝑤 ∈ 𝐴)
24	20, 23	impbid1 142	. 2 ⊢ (Rel 𝐴 → (∃𝑤 𝑤 ∈ 𝐴 ↔ ∃𝑦 𝑦 ∈ dom 𝐴))
25	2, 24	bitr3id 194	1 ⊢ (Rel 𝐴 → (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑦 𝑦 ∈ dom 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ⟨cop 3669  dom cdm 4718  Rel wrel 4723

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-xp 4724  df-rel 4725  df-dm 4728

This theorem is referenced by:  wlkm  16038