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

Theorem relresfld 6298
Description: Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.)
Assertion
Ref Expression
relresfld (Rel 𝑅 → (𝑅 𝑅) = 𝑅)

Proof of Theorem relresfld
StepHypRef Expression
1 relfld 6297 . . . 4 (Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))
21reseq2d 6000 . . 3 (Rel 𝑅 → (𝑅 𝑅) = (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)))
3 resundi 6014 . . 3 (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅))
4 eqtr 2758 . . . 4 (((𝑅 𝑅) = (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) ∧ (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅))) → (𝑅 𝑅) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)))
5 resss 6022 . . . . 5 (𝑅 ↾ ran 𝑅) ⊆ 𝑅
6 resdm 6046 . . . . 5 (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
7 ssequn2 4199 . . . . . 6 ((𝑅 ↾ ran 𝑅) ⊆ 𝑅 ↔ (𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅)
8 uneq1 4171 . . . . . . . . 9 ((𝑅 ↾ dom 𝑅) = 𝑅 → ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) = (𝑅 ∪ (𝑅 ↾ ran 𝑅)))
98eqeq2d 2746 . . . . . . . 8 ((𝑅 ↾ dom 𝑅) = 𝑅 → ((𝑅 𝑅) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) ↔ (𝑅 𝑅) = (𝑅 ∪ (𝑅 ↾ ran 𝑅))))
10 eqtr 2758 . . . . . . . . 9 (((𝑅 𝑅) = (𝑅 ∪ (𝑅 ↾ ran 𝑅)) ∧ (𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅) → (𝑅 𝑅) = 𝑅)
1110ex 412 . . . . . . . 8 ((𝑅 𝑅) = (𝑅 ∪ (𝑅 ↾ ran 𝑅)) → ((𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅 → (𝑅 𝑅) = 𝑅))
129, 11biimtrdi 253 . . . . . . 7 ((𝑅 ↾ dom 𝑅) = 𝑅 → ((𝑅 𝑅) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) → ((𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅 → (𝑅 𝑅) = 𝑅)))
1312com3r 87 . . . . . 6 ((𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅 → ((𝑅 ↾ dom 𝑅) = 𝑅 → ((𝑅 𝑅) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) → (𝑅 𝑅) = 𝑅)))
147, 13sylbi 217 . . . . 5 ((𝑅 ↾ ran 𝑅) ⊆ 𝑅 → ((𝑅 ↾ dom 𝑅) = 𝑅 → ((𝑅 𝑅) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) → (𝑅 𝑅) = 𝑅)))
155, 6, 14mpsyl 68 . . . 4 (Rel 𝑅 → ((𝑅 𝑅) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) → (𝑅 𝑅) = 𝑅))
164, 15syl5com 31 . . 3 (((𝑅 𝑅) = (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) ∧ (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅))) → (Rel 𝑅 → (𝑅 𝑅) = 𝑅))
172, 3, 16sylancl 586 . 2 (Rel 𝑅 → (Rel 𝑅 → (𝑅 𝑅) = 𝑅))
1817pm2.43i 52 1 (Rel 𝑅 → (𝑅 𝑅) = 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  cun 3961  wss 3963   cuni 4912  dom cdm 5689  ran crn 5690  cres 5691  Rel wrel 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701
This theorem is referenced by:  relcoi1  6300
  Copyright terms: Public domain W3C validator