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Theorem relresfld 6269
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 6268 . . . 4 (Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))
21reseq2d 5975 . . 3 (Rel 𝑅 → (𝑅 𝑅) = (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)))
3 resundi 5989 . . 3 (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅))
4 eqtr 2749 . . . 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 6000 . . . . 5 (𝑅 ↾ ran 𝑅) ⊆ 𝑅
6 resdm 6020 . . . . 5 (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
7 ssequn2 4178 . . . . . 6 ((𝑅 ↾ ran 𝑅) ⊆ 𝑅 ↔ (𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅)
8 uneq1 4151 . . . . . . . . 9 ((𝑅 ↾ dom 𝑅) = 𝑅 → ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) = (𝑅 ∪ (𝑅 ↾ ran 𝑅)))
98eqeq2d 2737 . . . . . . . 8 ((𝑅 ↾ dom 𝑅) = 𝑅 → ((𝑅 𝑅) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) ↔ (𝑅 𝑅) = (𝑅 ∪ (𝑅 ↾ ran 𝑅))))
10 eqtr 2749 . . . . . . . . 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 252 . . . . . . 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 216 . . . . 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 585 . 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 1533  cun 3941  wss 3943   cuni 4902  dom cdm 5669  ran crn 5670  cres 5671  Rel wrel 5674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681
This theorem is referenced by:  relcoi1  6271
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