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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 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) |
Colors of variables: wff set class |
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-bndl 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 |
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