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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 3473 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | vex 3473 | . . . . 5 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | opeldm 5904 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑥 ∈ dom 𝐴) |
5 | 2, 3 | opelrn 5939 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 ∈ ran 𝐴) |
6 | 4, 5 | opelxpd 5711 | . . 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 5784 | 1 ⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 ⊆ wss 3944 〈cop 4630 × cxp 5670 dom cdm 5672 ran crn 5673 Rel wrel 5677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-opab 5205 df-xp 5678 df-rel 5679 df-cnv 5680 df-dm 5682 df-rn 5683 |
This theorem is referenced by: resssxp 6268 cnvssrndm 6269 cossxp 6270 relrelss 6271 relfld 6273 fssxp 6745 oprabss 7521 cnvexg 7926 resfunexgALT 7945 cofunexg 7946 fnexALT 7948 funexw 7949 erssxp 8741 ttrclexg 9738 wunco 10748 trclublem 14966 trclubi 14967 trclub 14969 reltrclfv 14988 imasless 17513 sylow2a 19565 gsum2d 19918 znleval 21475 tsmsxp 24046 relfi 32377 fcnvgreu 32442 relssinxpdmrn 37757 trclubNEW 42972 trrelsuperreldg 43021 trrelsuperrel2dg 43024 |
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