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Theorem relfld 6295
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 6288 . . . 4 (Rel 𝑅𝑅 ⊆ (dom 𝑅 × ran 𝑅))
2 uniss 4915 . . . 4 (𝑅 ⊆ (dom 𝑅 × ran 𝑅) → 𝑅 (dom 𝑅 × ran 𝑅))
3 uniss 4915 . . . 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 5820 . . 3 (dom 𝑅 × ran 𝑅) ⊆ (dom 𝑅 ∪ ran 𝑅)
64, 5sstrdi 3996 . 2 (Rel 𝑅 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅))
7 dmrnssfld 5984 . . 3 (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅
87a1i 11 . 2 (Rel 𝑅 → (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅)
96, 8eqssd 4001 1 (Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  cun 3949  wss 3951   cuni 4907   × cxp 5683  dom cdm 5685  ran crn 5686  Rel wrel 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696
This theorem is referenced by:  relresfld  6296  unidmrn  6299  relcnvfld  6300  unixp  6302  relexp0  15062  relexpfld  15088  rtrclreclem4  15100  dfrtrcl2  15101  lefld  18637  fvmptiunrelexplb0da  43698
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