Theorem relssdmrn 6216

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 22	. 2 ⊢ (Rel 𝐴 → Rel 𝐴)
2		vex 3440	. . . . 5 ⊢ 𝑥 ∈ V
3		vex 3440	. . . . 5 ⊢ 𝑦 ∈ V
4	2, 3	opeldm 5846	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴)
5	2, 3	opelrn 5882	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 ∈ ran 𝐴)
6	4, 5	opelxpd 5653	. . 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 5727	1 ⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴))

Syntax hints:   → wi 4   ∈ wcel 2111   ⊆ wss 3897  ⟨cop 4579   × cxp 5612  dom cdm 5614  ran crn 5615  Rel wrel 5619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-xp 5620  df-rel 5621  df-cnv 5622  df-dm 5624  df-rn 5625

This theorem is referenced by:  resssxp  6217  cnvssrndm  6218  cossxp  6219  relrelss  6220  relfld  6222  fssxp  6678  oprabss  7454  cnvexg  7854  resfunexgALT  7880  cofunexg  7881  fnexALT  7883  funexw  7884  erssxp  8645  ttrclexg  9613  wunco  10624  trclublem  14902  trclubi  14903  trclub  14905  reltrclfv  14924  imasless  17444  sylow2a  19531  gsum2d  19884  znleval  21491  tsmsxp  24070  relfi  32582  fcnvgreu  32655  elrgspnsubrunlem2  33215  relssinxpdmrn  38380  trclubNEW  43711  trrelsuperreldg  43760  trrelsuperrel2dg  43763  relwf  45059