Theorem relssdmrn 6270

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.)

Assertion

Ref	Expression
relssdmrn	⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴))

Proof of Theorem relssdmrn

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 23	. 2 ⊢ (Rel 𝐴 → Rel 𝐴)
2		vex 3457	. . . . 5 ⊢ 𝑥 ∈ V
3		vex 3457	. . . . 5 ⊢ 𝑦 ∈ V
4	2, 3	opeldm 5895	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴)
5	2, 3	opelrn 5931	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 ∈ ran 𝐴)
6	4, 5	opelxpd 5698	. . 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 5772	1 ⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴))

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   ⊆ wss 3902  ⟨cop 4593   × cxp 5657  dom cdm 5659  ran crn 5660  Rel wrel 5664

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-cnv 5667  df-dm 5669  df-rn 5670

This theorem is used by:  resssxp  6271  cnvssrndm  6272  cossxp  6273  relrelss  6274  relfld  6276  fssxp  6734  oprabss  7524  cnvexg  7924  resfunexgALT  7948  cofunexg  7949  fnexALT  7951  funexw  7952  erssxp  8723  ttrclexg  9705  wunco  10745  trclublem  15070  trclubi  15071  trclub  15073  reltrclfv  15092  imasless  17630  sylow2a  19747  gsum2d  20100  znleval  21768  tsmsxp  24382  relfi  33062  fcnvgreu  33132  elrgspnsubrunlem2  33675  relssinxpdmrn  39084  trclubNEW  44446  trrelsuperreldg  44495  trrelsuperrel2dg  44498  relwf  45777