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Mirrors > Home > ILE Home > Th. List > releldmb | GIF version |
Description: Membership in a domain. (Contributed by Mario Carneiro, 5-Nov-2015.) |
Ref | Expression |
---|---|
releldmb | ⊢ (Rel 𝑅 → (A ∈ dom 𝑅 ↔ ∃x A𝑅x)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldmg 4473 | . . 3 ⊢ (A ∈ dom 𝑅 → (A ∈ dom 𝑅 ↔ ∃x A𝑅x)) | |
2 | 1 | ibi 165 | . 2 ⊢ (A ∈ dom 𝑅 → ∃x A𝑅x) |
3 | releldm 4512 | . . . 4 ⊢ ((Rel 𝑅 ∧ A𝑅x) → A ∈ dom 𝑅) | |
4 | 3 | ex 108 | . . 3 ⊢ (Rel 𝑅 → (A𝑅x → A ∈ dom 𝑅)) |
5 | 4 | exlimdv 1697 | . 2 ⊢ (Rel 𝑅 → (∃x A𝑅x → A ∈ dom 𝑅)) |
6 | 2, 5 | impbid2 131 | 1 ⊢ (Rel 𝑅 → (A ∈ dom 𝑅 ↔ ∃x A𝑅x)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∃wex 1378 ∈ wcel 1390 class class class wbr 3755 dom cdm 4288 Rel wrel 4293 |
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-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 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-rel 4295 df-dm 4298 |
This theorem is referenced by: (None) |
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