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Mirrors > Home > ILE Home > Th. List > relssdmrn | GIF version |
Description: A relation is included in the cross product of its domain and range. Exercise 4.12(t) of [Mendelson] p. 235. (Contributed by NM, 3-Aug-1994.) |
Ref | Expression |
---|---|
relssdmrn | ⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . 2 ⊢ (Rel 𝐴 → Rel 𝐴) | |
2 | 19.8a 1569 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) | |
3 | 19.8a 1569 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴) | |
4 | opelxp 4564 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (dom 𝐴 × ran 𝐴) ↔ (𝑥 ∈ dom 𝐴 ∧ 𝑦 ∈ ran 𝐴)) | |
5 | vex 2684 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
6 | 5 | eldm2 4732 | . . . . . 6 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) |
7 | vex 2684 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
8 | 7 | elrn2 4776 | . . . . . 6 ⊢ (𝑦 ∈ ran 𝐴 ↔ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴) |
9 | 6, 8 | anbi12i 455 | . . . . 5 ⊢ ((𝑥 ∈ dom 𝐴 ∧ 𝑦 ∈ ran 𝐴) ↔ (∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴 ∧ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴)) |
10 | 4, 9 | bitri 183 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (dom 𝐴 × ran 𝐴) ↔ (∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴 ∧ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴)) |
11 | 2, 3, 10 | sylanbrc 413 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ (dom 𝐴 × ran 𝐴)) |
12 | 11 | a1i 9 | . 2 ⊢ (Rel 𝐴 → (〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ (dom 𝐴 × ran 𝐴))) |
13 | 1, 12 | relssdv 4626 | 1 ⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∃wex 1468 ∈ wcel 1480 ⊆ wss 3066 〈cop 3525 × cxp 4532 dom cdm 4534 ran crn 4535 Rel wrel 4539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-xp 4540 df-rel 4541 df-cnv 4542 df-dm 4544 df-rn 4545 |
This theorem is referenced by: cnvssrndm 5055 cossxp 5056 relrelss 5060 relfld 5062 cnvexg 5071 fssxp 5285 oprabss 5850 resfunexgALT 6001 cofunexg 6002 fnexALT 6004 erssxp 6445 |
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