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Theorem relssdmrn 4784
Description: A relation is included in the cross product of its domain and range. Exercise 4.12(t) of [Mendelson] p. 235. (Contributed by NM, 3-Aug-1994.)
Assertion
Ref Expression
relssdmrn (Rel AA ⊆ (dom A × ran A))

Proof of Theorem relssdmrn
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 19 . 2 (Rel A → Rel A)
2 19.8a 1479 . . . 4 (⟨x, y Ayx, y A)
3 19.8a 1479 . . . 4 (⟨x, y Axx, y A)
4 opelxp 4317 . . . . 5 (⟨x, y (dom A × ran A) ↔ (x dom A y ran A))
5 vex 2554 . . . . . . 7 x V
65eldm2 4476 . . . . . 6 (x dom Ayx, y A)
7 vex 2554 . . . . . . 7 y V
87elrn2 4519 . . . . . 6 (y ran Axx, y A)
96, 8anbi12i 433 . . . . 5 ((x dom A y ran A) ↔ (yx, y A xx, y A))
104, 9bitri 173 . . . 4 (⟨x, y (dom A × ran A) ↔ (yx, y A xx, y A))
112, 3, 10sylanbrc 394 . . 3 (⟨x, y A → ⟨x, y (dom A × ran A))
1211a1i 9 . 2 (Rel A → (⟨x, y A → ⟨x, y (dom A × ran A)))
131, 12relssdv 4375 1 (Rel AA ⊆ (dom A × ran A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wex 1378   wcel 1390  wss 2911  cop 3370   × cxp 4286  dom cdm 4288  ran crn 4289  Rel wrel 4293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299
This theorem is referenced by:  cnvssrndm  4785  cossxp  4786  relrelss  4787  relfld  4789  cnvexg  4798  fssxp  5001  oprabss  5532  resfunexgALT  5679  cofunexg  5680  fnexALT  5682  erssxp  6065
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