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Theorem relfld 5699
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 5694 . . . 4 (Rel 𝑅𝑅 ⊆ (dom 𝑅 × ran 𝑅))
2 uniss 4490 . . . 4 (𝑅 ⊆ (dom 𝑅 × ran 𝑅) → 𝑅 (dom 𝑅 × ran 𝑅))
3 uniss 4490 . . . 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 5267 . . 3 (dom 𝑅 × ran 𝑅) ⊆ (dom 𝑅 ∪ ran 𝑅)
64, 5syl6ss 3648 . 2 (Rel 𝑅 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅))
7 dmrnssfld 5416 . . 3 (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅
87a1i 11 . 2 (Rel 𝑅 → (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅)
96, 8eqssd 3653 1 (Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  cun 3605  wss 3607   cuni 4468   × cxp 5141  dom cdm 5143  ran crn 5144  Rel wrel 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-dm 5153  df-rn 5154
This theorem is referenced by:  relresfld  5700  unidmrn  5703  relcnvfld  5704  unixp  5706  relexp0  13807  relexpfld  13833  rtrclreclem4  13845  dfrtrcl2  13846  lefld  17273  fvmptiunrelexplb0da  38294
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