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Theorem relfld 4846
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 4841 . . . 4 (Rel 𝑅𝑅 ⊆ (dom 𝑅 × ran 𝑅))
2 uniss 3601 . . . 4 (𝑅 ⊆ (dom 𝑅 × ran 𝑅) → 𝑅 (dom 𝑅 × ran 𝑅))
3 uniss 3601 . . . 4 ( 𝑅 (dom 𝑅 × ran 𝑅) → 𝑅 (dom 𝑅 × ran 𝑅))
41, 2, 33syl 17 . . 3 (Rel 𝑅 𝑅 (dom 𝑅 × ran 𝑅))
5 unixpss 4451 . . 3 (dom 𝑅 × ran 𝑅) ⊆ (dom 𝑅 ∪ ran 𝑅)
64, 5syl6ss 2957 . 2 (Rel 𝑅 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅))
7 dmrnssfld 4595 . . 3 (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅
87a1i 9 . 2 (Rel 𝑅 → (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅)
96, 8eqssd 2962 1 (Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1243  cun 2915  wss 2917   cuni 3580   × cxp 4343  dom cdm 4345  ran crn 4346  Rel wrel 4350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356
This theorem is referenced by:  relresfld  4847  relcoi1  4849  unidmrn  4850  relcnvfld  4851  unixpm  4853
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