MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  relfld Structured version   Visualization version   GIF version

Theorem relfld 6281
Description: The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.)
Assertion
Ref Expression
relfld (Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))

Proof of Theorem relfld
StepHypRef Expression
1 relssdmrn 6274 . . . 4 (Rel 𝑅𝑅 ⊆ (dom 𝑅 × ran 𝑅))
2 uniss 4917 . . . 4 (𝑅 ⊆ (dom 𝑅 × ran 𝑅) → 𝑅 (dom 𝑅 × ran 𝑅))
3 uniss 4917 . . . 4 ( 𝑅 (dom 𝑅 × ran 𝑅) → 𝑅 (dom 𝑅 × ran 𝑅))
41, 2, 33syl 18 . . 3 (Rel 𝑅 𝑅 (dom 𝑅 × ran 𝑅))
5 unixpss 5812 . . 3 (dom 𝑅 × ran 𝑅) ⊆ (dom 𝑅 ∪ ran 𝑅)
64, 5sstrdi 3989 . 2 (Rel 𝑅 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅))
7 dmrnssfld 5973 . . 3 (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅
87a1i 11 . 2 (Rel 𝑅 → (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅)
96, 8eqssd 3994 1 (Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  cun 3942  wss 3944   cuni 4909   × cxp 5676  dom cdm 5678  ran crn 5679  Rel wrel 5683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5684  df-rel 5685  df-cnv 5686  df-dm 5688  df-rn 5689
This theorem is referenced by:  relresfld  6282  unidmrn  6285  relcnvfld  6286  unixp  6288  relexp0  15006  relexpfld  15032  rtrclreclem4  15044  dfrtrcl2  15045  lefld  18587  fvmptiunrelexplb0da  43254
  Copyright terms: Public domain W3C validator