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Theorem relfld 6128
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 6123 . . . 4 (Rel 𝑅𝑅 ⊆ (dom 𝑅 × ran 𝑅))
2 uniss 4848 . . . 4 (𝑅 ⊆ (dom 𝑅 × ran 𝑅) → 𝑅 (dom 𝑅 × ran 𝑅))
3 uniss 4848 . . . 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 5685 . . 3 (dom 𝑅 × ran 𝑅) ⊆ (dom 𝑅 ∪ ran 𝑅)
64, 5sstrdi 3981 . 2 (Rel 𝑅 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅))
7 dmrnssfld 5843 . . 3 (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅
87a1i 11 . 2 (Rel 𝑅 → (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅)
96, 8eqssd 3986 1 (Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  cun 3936  wss 3938   cuni 4840   × cxp 5555  dom cdm 5557  ran crn 5558  Rel wrel 5562
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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-xp 5563  df-rel 5564  df-cnv 5565  df-dm 5567  df-rn 5568
This theorem is referenced by:  relresfld  6129  unidmrn  6132  relcnvfld  6133  unixp  6135  relexp0  14384  relexpfld  14410  rtrclreclem4  14422  dfrtrcl2  14423  lefld  17838  fvmptiunrelexplb0da  40037
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