Theorem ecdmn0m 6401

Description: A representative of an inhabited equivalence class belongs to the domain of the equivalence relation. (Contributed by Jim Kingdon, 21-Aug-2019.)

Assertion

Ref	Expression
ecdmn0m	⊢ (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅)

Distinct variable groups:   𝑥,𝑅   𝑥,𝐴

Proof of Theorem ecdmn0m

Step	Hyp	Ref	Expression
1		elex 2652	. 2 ⊢ (𝐴 ∈ dom 𝑅 → 𝐴 ∈ V)
2		ecexr 6364	. . 3 ⊢ (𝑥 ∈ [𝐴]𝑅 → 𝐴 ∈ V)
3	2	exlimiv 1545	. 2 ⊢ (∃𝑥 𝑥 ∈ [𝐴]𝑅 → 𝐴 ∈ V)
4		eldmg 4672	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
5		vex 2644	. . . . 5 ⊢ 𝑥 ∈ V
6		elecg 6397	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ V) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
7	5, 6	mpan 418	. . . 4 ⊢ (𝐴 ∈ V → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
8	7	exbidv 1764	. . 3 ⊢ (𝐴 ∈ V → (∃𝑥 𝑥 ∈ [𝐴]𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
9	4, 8	bitr4d 190	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅))
10	1, 3, 9	pm5.21nii 661	1 ⊢ (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅)

Syntax hints:   ↔ wb 104  ∃wex 1436   ∈ wcel 1448  Vcvv 2641   class class class wbr 3875  dom cdm 4477  [cec 6357

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876  df-opab 3930  df-xp 4483  df-cnv 4485  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-ec 6361

This theorem is referenced by:  ereldm  6402  elqsn0m  6427  ecelqsdm  6429