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Mirrors > Home > MPE Home > Th. List > releldmb | Structured version Visualization version 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 5761 | . . 3 ⊢ (𝐴 ∈ dom 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥)) | |
2 | 1 | ibi 269 | . 2 ⊢ (𝐴 ∈ dom 𝑅 → ∃𝑥 𝐴𝑅𝑥) |
3 | releldm 5808 | . . . 4 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝑥) → 𝐴 ∈ dom 𝑅) | |
4 | 3 | ex 415 | . . 3 ⊢ (Rel 𝑅 → (𝐴𝑅𝑥 → 𝐴 ∈ dom 𝑅)) |
5 | 4 | exlimdv 1930 | . 2 ⊢ (Rel 𝑅 → (∃𝑥 𝐴𝑅𝑥 → 𝐴 ∈ dom 𝑅)) |
6 | 2, 5 | impbid2 228 | 1 ⊢ (Rel 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∃wex 1776 ∈ wcel 2110 class class class wbr 5058 dom cdm 5549 Rel wrel 5554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-xp 5555 df-rel 5556 df-dm 5559 |
This theorem is referenced by: eqvrelref 35839 |
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