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| Mirrors > Home > MPE Home > Th. List > relssdmrnOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of relssdmrn 6288 as of 23-Dec-2024. (Contributed by NM, 3-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| relssdmrnOLD | ⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | id 22 | . 2 ⊢ (Rel 𝐴 → Rel 𝐴) | |
| 2 | 19.8a 2181 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) | |
| 3 | 19.8a 2181 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴) | |
| 4 | opelxp 5721 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (dom 𝐴 × ran 𝐴) ↔ (𝑥 ∈ dom 𝐴 ∧ 𝑦 ∈ ran 𝐴)) | |
| 5 | vex 3484 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
| 6 | 5 | eldm2 5912 | . . . . . 6 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) |
| 7 | vex 3484 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
| 8 | 7 | elrn2 5903 | . . . . . 6 ⊢ (𝑦 ∈ ran 𝐴 ↔ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴) |
| 9 | 6, 8 | anbi12i 628 | . . . . 5 ⊢ ((𝑥 ∈ dom 𝐴 ∧ 𝑦 ∈ ran 𝐴) ↔ (∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴 ∧ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴)) |
| 10 | 4, 9 | bitri 275 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (dom 𝐴 × ran 𝐴) ↔ (∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴 ∧ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴)) |
| 11 | 2, 3, 10 | sylanbrc 583 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ (dom 𝐴 × ran 𝐴)) |
| 12 | 11 | a1i 11 | . 2 ⊢ (Rel 𝐴 → (〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ (dom 𝐴 × ran 𝐴))) |
| 13 | 1, 12 | relssdv 5798 | 1 ⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∃wex 1779 ∈ wcel 2108 ⊆ wss 3951 〈cop 4632 × cxp 5683 dom cdm 5685 ran crn 5686 Rel wrel 5690 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 |
| This theorem is referenced by: (None) |
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