Theorem ecdmn0m 6059

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 2543	. 2 ⊢ (A ∈ dom 𝑅 → A ∈ V)
2		ecexr 6022	. . 3 ⊢ (x ∈ [A]𝑅 → A ∈ V)
3	2	exlimiv 1471	. 2 ⊢ (∃x x ∈ [A]𝑅 → A ∈ V)
4		eldmg 4457	. . 3 ⊢ (A ∈ V → (A ∈ dom 𝑅 ↔ ∃x A𝑅x))
5		vex 2538	. . . . 5 ⊢ x ∈ V
6		elecg 6055	. . . . 5 ⊢ ((x ∈ V ∧ A ∈ V) → (x ∈ [A]𝑅 ↔ A𝑅x))
7	5, 6	mpan 402	. . . 4 ⊢ (A ∈ V → (x ∈ [A]𝑅 ↔ A𝑅x))
8	7	exbidv 1688	. . 3 ⊢ (A ∈ V → (∃x x ∈ [A]𝑅 ↔ ∃x A𝑅x))
9	4, 8	bitr4d 180	. 2 ⊢ (A ∈ V → (A ∈ dom 𝑅 ↔ ∃x x ∈ [A]𝑅))
10	1, 3, 9	pm5.21nii 607	1 ⊢ (A ∈ dom 𝑅 ↔ ∃x x ∈ [A]𝑅)

Syntax hints:   ↔ wb 98  ∃wex 1362   ∈ wcel 1374  Vcvv 2535   class class class wbr 3738  dom cdm 4272  [cec 6015

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-xp 4278  df-cnv 4280  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-ec 6019

This theorem is referenced by:  ereldm  6060  elqsn0m  6085  ecelqsdm  6087