Theorem ecdmn0m 6851

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 2833	. 2 ⊢ (𝐴 ∈ dom 𝑅 → 𝐴 ∈ V)
2		ecexr 6812	. . 3 ⊢ (𝑥 ∈ [𝐴]𝑅 → 𝐴 ∈ V)
3	2	exlimiv 1651	. 2 ⊢ (∃𝑥 𝑥 ∈ [𝐴]𝑅 → 𝐴 ∈ V)
4		eldmg 4976	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
5		vex 2824	. . . . 5 ⊢ 𝑥 ∈ V
6		elecg 6847	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ V) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
7	5, 6	mpan 428	. . . 4 ⊢ (𝐴 ∈ V → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
8	7	exbidv 1878	. . 3 ⊢ (𝐴 ∈ V → (∃𝑥 𝑥 ∈ [𝐴]𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
9	4, 8	bitr4d 191	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅))
10	1, 3, 9	pm5.21nii 716	1 ⊢ (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅)

This proof depends on syntax axioms:   ↔ wb 105  ∃wex 1545   ∈ wcel 2209  Vcvv 2821   class class class wbr 4130  dom cdm 4774  [cec 6805

This proof depends on 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 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  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-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-xp 4780  df-cnv 4782  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-ec 6809

This theorem is used by:  ereldm  6852  elqsn0m  6877  ecelqsdm  6879  divsfval  13649