Theorem ecdmn0m 6084

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 2560	. 2 ⊢ (A ∈ dom 𝑅 → A ∈ V)
2		ecexr 6047	. . 3 ⊢ (x ∈ [A]𝑅 → A ∈ V)
3	2	exlimiv 1486	. 2 ⊢ (∃x x ∈ [A]𝑅 → A ∈ V)
4		eldmg 4473	. . 3 ⊢ (A ∈ V → (A ∈ dom 𝑅 ↔ ∃x A𝑅x))
5		vex 2554	. . . . 5 ⊢ x ∈ V
6		elecg 6080	. . . . 5 ⊢ ((x ∈ V ∧ A ∈ V) → (x ∈ [A]𝑅 ↔ A𝑅x))
7	5, 6	mpan 400	. . . 4 ⊢ (A ∈ V → (x ∈ [A]𝑅 ↔ A𝑅x))
8	7	exbidv 1703	. . 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 619	1 ⊢ (A ∈ dom 𝑅 ↔ ∃x x ∈ [A]𝑅)

Syntax hints:   ↔ wb 98  ∃wex 1378   ∈ wcel 1390  Vcvv 2551   class class class wbr 3755  dom cdm 4288  [cec 6040

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-ec 6044

This theorem is referenced by:  ereldm  6085  elqsn0m  6110  ecelqsdm  6112