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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 2692 | . 2 ⊢ (𝐴 ∈ dom 𝑅 → 𝐴 ∈ V) | |
2 | ecexr 6427 | . . 3 ⊢ (𝑥 ∈ [𝐴]𝑅 → 𝐴 ∈ V) | |
3 | 2 | exlimiv 1577 | . 2 ⊢ (∃𝑥 𝑥 ∈ [𝐴]𝑅 → 𝐴 ∈ V) |
4 | eldmg 4729 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥)) | |
5 | vex 2684 | . . . . 5 ⊢ 𝑥 ∈ V | |
6 | elecg 6460 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ V) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥)) | |
7 | 5, 6 | mpan 420 | . . . 4 ⊢ (𝐴 ∈ V → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥)) |
8 | 7 | exbidv 1797 | . . 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 693 | 1 ⊢ (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∃wex 1468 ∈ wcel 1480 Vcvv 2681 class class class wbr 3924 dom cdm 4534 [cec 6420 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-xp 4540 df-cnv 4542 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-ec 6424 |
This theorem is referenced by: ereldm 6465 elqsn0m 6490 ecelqsdm 6492 |
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