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Mirrors > Home > MPE Home > Th. List > relssdmrn | Structured version Visualization version GIF version |
Description: A relation is included in the Cartesian 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 22 | . 2 ⊢ (Rel 𝐴 → Rel 𝐴) | |
2 | 19.8a 2039 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) | |
3 | 19.8a 2039 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴) | |
4 | opelxp 5070 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (dom 𝐴 × ran 𝐴) ↔ (𝑥 ∈ dom 𝐴 ∧ 𝑦 ∈ ran 𝐴)) | |
5 | vex 3176 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
6 | 5 | eldm2 5244 | . . . . . 6 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) |
7 | vex 3176 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
8 | 7 | elrn2 5286 | . . . . . 6 ⊢ (𝑦 ∈ ran 𝐴 ↔ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴) |
9 | 6, 8 | anbi12i 729 | . . . . 5 ⊢ ((𝑥 ∈ dom 𝐴 ∧ 𝑦 ∈ ran 𝐴) ↔ (∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴 ∧ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴)) |
10 | 4, 9 | bitri 263 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (dom 𝐴 × ran 𝐴) ↔ (∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴 ∧ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴)) |
11 | 2, 3, 10 | sylanbrc 695 | . . 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 5135 | 1 ⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∃wex 1695 ∈ wcel 1977 ⊆ wss 3540 〈cop 4131 × cxp 5036 dom cdm 5038 ran crn 5039 Rel wrel 5043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 |
This theorem is referenced by: cnvssrndm 5574 cossxp 5575 relrelss 5576 relfld 5578 fssxp 5973 oprabss 6644 cnvexg 7005 resfunexgALT 7022 cofunexg 7023 fnexALT 7025 erssxp 7652 wunco 9434 trclublem 13582 trclubi 13583 trclubiOLD 13584 trclub 13587 reltrclfv 13606 imasless 16023 sylow2a 17857 gsum2d 18194 znleval 19722 tsmsxp 21768 relfi 28797 idssxp 28811 fcnvgreu 28855 rtrclex 36943 trclubNEW 36945 rtrclexi 36947 trrelsuperreldg 36979 trrelsuperrel2dg 36982 rp-imass 37085 idhe 37101 |
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