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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.) (Proof shortened by SN, 23-Dec-2024.) |
| Ref | Expression |
|---|---|
| relssdmrn | ⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | id 22 | . 2 ⊢ (Rel 𝐴 → Rel 𝐴) | |
| 2 | vex 3484 | . . . . 5 ⊢ 𝑥 ∈ V | |
| 3 | vex 3484 | . . . . 5 ⊢ 𝑦 ∈ V | |
| 4 | 2, 3 | opeldm 5918 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑥 ∈ dom 𝐴) |
| 5 | 2, 3 | opelrn 5954 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 ∈ ran 𝐴) |
| 6 | 4, 5 | opelxpd 5724 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ (dom 𝐴 × ran 𝐴)) |
| 7 | 6 | a1i 11 | . 2 ⊢ (Rel 𝐴 → (〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ (dom 𝐴 × ran 𝐴))) |
| 8 | 1, 7 | relssdv 5798 | 1 ⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ 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-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: resssxp 6290 cnvssrndm 6291 cossxp 6292 relrelss 6293 relfld 6295 fssxp 6763 oprabss 7541 cnvexg 7946 resfunexgALT 7972 cofunexg 7973 fnexALT 7975 funexw 7976 erssxp 8768 ttrclexg 9763 wunco 10773 trclublem 15034 trclubi 15035 trclub 15037 reltrclfv 15056 imasless 17585 sylow2a 19637 gsum2d 19990 znleval 21573 tsmsxp 24163 relfi 32615 fcnvgreu 32683 elrgspnsubrunlem2 33252 relssinxpdmrn 38350 trclubNEW 43632 trrelsuperreldg 43681 trrelsuperrel2dg 43684 relwf 44984 |
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