Theorem relssdmrn 6225

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 3434	. . . . 5 ⊢ 𝑥 ∈ V
3		vex 3434	. . . . 5 ⊢ 𝑦 ∈ V
4	2, 3	opeldm 5854	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴)
5	2, 3	opelrn 5890	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 ∈ ran 𝐴)
6	4, 5	opelxpd 5661	. . 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 5735	1 ⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴))

Syntax hints:   → wi 4   ∈ wcel 2114   ⊆ wss 3890  ⟨cop 4574   × cxp 5620  dom cdm 5622  ran crn 5623  Rel wrel 5627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5628  df-rel 5629  df-cnv 5630  df-dm 5632  df-rn 5633

This theorem is referenced by:  resssxp  6226  cnvssrndm  6227  cossxp  6228  relrelss  6229  relfld  6231  fssxp  6687  oprabss  7466  cnvexg  7866  resfunexgALT  7892  cofunexg  7893  fnexALT  7895  funexw  7896  erssxp  8658  ttrclexg  9633  wunco  10645  trclublem  14946  trclubi  14947  trclub  14949  reltrclfv  14968  imasless  17493  sylow2a  19583  gsum2d  19936  znleval  21542  tsmsxp  24129  relfi  32692  fcnvgreu  32765  elrgspnsubrunlem2  33329  relssinxpdmrn  38681  trclubNEW  44061  trrelsuperreldg  44110  trrelsuperrel2dg  44113  relwf  45409