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Mirrors > Home > ILE Home > Th. List > ecdmn0m | GIF version |
Description: A representative of an inhabited equivalence class belongs to the domain of the equivalence relation. (Contributed by Jim Kingdon, 21-Aug-2019.) |
Ref | Expression |
---|---|
ecdmn0m | ⊢ (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅) |
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) |
Colors of variables: wff set class |
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 |
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