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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 4821 | . . 3 ⊢ (𝐴 ∈ dom 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥)) | |
2 | 1 | ibi 176 | . 2 ⊢ (𝐴 ∈ dom 𝑅 → ∃𝑥 𝐴𝑅𝑥) |
3 | releldm 4861 | . . . 4 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝑥) → 𝐴 ∈ dom 𝑅) | |
4 | 3 | ex 115 | . . 3 ⊢ (Rel 𝑅 → (𝐴𝑅𝑥 → 𝐴 ∈ dom 𝑅)) |
5 | 4 | exlimdv 1819 | . 2 ⊢ (Rel 𝑅 → (∃𝑥 𝐴𝑅𝑥 → 𝐴 ∈ dom 𝑅)) |
6 | 2, 5 | impbid2 143 | 1 ⊢ (Rel 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∃wex 1492 ∈ wcel 2148 class class class wbr 4002 dom cdm 4625 Rel wrel 4630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-br 4003 df-opab 4064 df-xp 4631 df-rel 4632 df-dm 4635 |
This theorem is referenced by: (None) |
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