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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 2052 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) | |
3 | 19.8a 2052 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴) | |
4 | opelxp 5146 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (dom 𝐴 × ran 𝐴) ↔ (𝑥 ∈ dom 𝐴 ∧ 𝑦 ∈ ran 𝐴)) | |
5 | vex 3203 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
6 | 5 | eldm2 5322 | . . . . . 6 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) |
7 | vex 3203 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
8 | 7 | elrn2 5365 | . . . . . 6 ⊢ (𝑦 ∈ ran 𝐴 ↔ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴) |
9 | 6, 8 | anbi12i 733 | . . . . 5 ⊢ ((𝑥 ∈ dom 𝐴 ∧ 𝑦 ∈ ran 𝐴) ↔ (∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴 ∧ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴)) |
10 | 4, 9 | bitri 264 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (dom 𝐴 × ran 𝐴) ↔ (∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴 ∧ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴)) |
11 | 2, 3, 10 | sylanbrc 698 | . . 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 5212 | 1 ⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∃wex 1704 ∈ wcel 1990 ⊆ wss 3574 〈cop 4183 × cxp 5112 dom cdm 5114 ran crn 5115 Rel wrel 5119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-dm 5124 df-rn 5125 |
This theorem is referenced by: cnvssrndm 5657 cossxp 5658 relrelss 5659 relfld 5661 idssxp 6009 fssxp 6060 oprabss 6746 cnvexg 7112 resfunexgALT 7129 cofunexg 7130 fnexALT 7132 erssxp 7765 wunco 9555 trclublem 13734 trclubi 13735 trclubiOLD 13736 trclub 13739 reltrclfv 13758 imasless 16200 sylow2a 18034 gsum2d 18371 znleval 19903 tsmsxp 21958 relfi 29415 fcnvgreu 29472 rtrclex 37924 trclubNEW 37926 rtrclexi 37928 trrelsuperreldg 37960 trrelsuperrel2dg 37963 rp-imass 38065 idhe 38081 |
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