Theorem reldmm 4995

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 2299	. . 3 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
2	1	cbvexv 1974	. 2 ⊢ (∃𝑤 𝑤 ∈ 𝐴 ↔ ∃𝑥 𝑥 ∈ 𝐴)
3		elrel 4872	. . . . . . . 8 ⊢ ((Rel 𝐴 ∧ 𝑤 ∈ 𝐴) → ∃𝑦∃𝑧 𝑤 = ⟨𝑦, 𝑧⟩)
4		eleq1 2301	. . . . . . . . . . . 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 1653	. . . . . . . . . 10 ⊢ (∃𝑧 𝑤 = ⟨𝑦, 𝑧⟩ → ∃𝑧(𝑤 ∈ 𝐴 → ⟨𝑦, 𝑧⟩ ∈ 𝐴))
7		nfv 1581	. . . . . . . . . . 11 ⊢ Ⅎ𝑧 𝑤 ∈ 𝐴
8	7	19.37-1 1726	. . . . . . . . . 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 1653	. . . . . . . 8 ⊢ (∃𝑦∃𝑧 𝑤 = ⟨𝑦, 𝑧⟩ → ∃𝑦(𝑤 ∈ 𝐴 → ∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝐴))
11		nfv 1581	. . . . . . . . 9 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐴
12	11	19.37-1 1726	. . . . . . . 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 1333	. . . . . 6 ⊢ ((Rel 𝐴 ∧ 𝑤 ∈ 𝐴) → ∃𝑦∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝐴)
15		vex 2824	. . . . . . . 8 ⊢ 𝑦 ∈ V
16	15	eldm2 4974	. . . . . . 7 ⊢ (𝑦 ∈ dom 𝐴 ↔ ∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝐴)
17	16	exbii 1658	. . . . . 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 1872	. . 3 ⊢ (Rel 𝐴 → (∃𝑤 𝑤 ∈ 𝐴 → ∃𝑦 𝑦 ∈ dom 𝐴))
21		elex2 2838	. . . . 5 ⊢ (⟨𝑦, 𝑧⟩ ∈ 𝐴 → ∃𝑤 𝑤 ∈ 𝐴)
22	21	exlimivv 1952	. . . 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 1402  ∃wex 1545   ∈ wcel 2209  ⟨cop 3708  dom cdm 4769  Rel wrel 4774

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 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 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-dm 4779

This theorem is referenced by:  wlkm  16494  eulerpathum  16636