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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 2566 | . 2 ⊢ (𝐴 ∈ dom 𝑅 → 𝐴 ∈ V) | |
2 | ecexr 6111 | . . 3 ⊢ (𝑥 ∈ [𝐴]𝑅 → 𝐴 ∈ V) | |
3 | 2 | exlimiv 1489 | . 2 ⊢ (∃𝑥 𝑥 ∈ [𝐴]𝑅 → 𝐴 ∈ V) |
4 | eldmg 4530 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥)) | |
5 | vex 2560 | . . . . 5 ⊢ 𝑥 ∈ V | |
6 | elecg 6144 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ V) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥)) | |
7 | 5, 6 | mpan 400 | . . . 4 ⊢ (𝐴 ∈ V → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥)) |
8 | 7 | exbidv 1706 | . . 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 620 | 1 ⊢ (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 ∃wex 1381 ∈ wcel 1393 Vcvv 2557 class class class wbr 3764 dom cdm 4345 [cec 6104 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-xp 4351 df-cnv 4353 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-ec 6108 |
This theorem is referenced by: ereldm 6149 elqsn0m 6174 ecelqsdm 6176 |
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