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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 2216 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) | |
3 | 19.8a 2216 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴) | |
4 | opelxp 5348 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (dom 𝐴 × ran 𝐴) ↔ (𝑥 ∈ dom 𝐴 ∧ 𝑦 ∈ ran 𝐴)) | |
5 | vex 3388 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
6 | 5 | eldm2 5525 | . . . . . 6 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴) |
7 | vex 3388 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
8 | 7 | elrn2 5569 | . . . . . 6 ⊢ (𝑦 ∈ ran 𝐴 ↔ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴) |
9 | 6, 8 | anbi12i 621 | . . . . 5 ⊢ ((𝑥 ∈ dom 𝐴 ∧ 𝑦 ∈ ran 𝐴) ↔ (∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴 ∧ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴)) |
10 | 4, 9 | bitri 267 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (dom 𝐴 × ran 𝐴) ↔ (∃𝑦〈𝑥, 𝑦〉 ∈ 𝐴 ∧ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴)) |
11 | 2, 3, 10 | sylanbrc 579 | . . 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 5416 | 1 ⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∃wex 1875 ∈ wcel 2157 ⊆ wss 3769 〈cop 4374 × cxp 5310 dom cdm 5312 ran crn 5313 Rel wrel 5317 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-br 4844 df-opab 4906 df-xp 5318 df-rel 5319 df-cnv 5320 df-dm 5322 df-rn 5323 |
This theorem is referenced by: cnvssrndm 5876 cossxp 5877 relrelss 5878 relfld 5880 idssxpOLD 6220 fssxp 6275 oprabss 6980 cnvexg 7347 resfunexgALT 7364 cofunexg 7365 fnexALT 7367 erssxp 8005 wunco 9843 trclublem 14077 trclubi 14078 trclub 14080 reltrclfv 14099 imasless 16515 sylow2a 18347 gsum2d 18686 znleval 20224 tsmsxp 22286 relfi 29932 fcnvgreu 29990 rtrclex 38707 trclubNEW 38709 rtrclexi 38711 trrelsuperreldg 38743 trrelsuperrel2dg 38746 rp-imass 38847 idhe 38863 |
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