Theorem relresfld 6277

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

Step	Hyp	Ref	Expression
1		relfld 6276	. . . 4 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))
2	1	reseq2d 5977	. . 3 ⊢ (Rel 𝑅 → (𝑅 ↾ ∪ ∪ 𝑅) = (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)))
3		resundi 5991	. . 3 ⊢ (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅))
4		eqtr 2782	. . . 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 5999	. . . . 5 ⊢ (𝑅 ↾ ran 𝑅) ⊆ 𝑅
6		resdm 6024	. . . . 5 ⊢ (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
7		ssequn2 4141	. . . . . 6 ⊢ ((𝑅 ↾ ran 𝑅) ⊆ 𝑅 ↔ (𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅)
8		uneq1 4114	. . . . . . . . 9 ⊢ ((𝑅 ↾ dom 𝑅) = 𝑅 → ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) = (𝑅 ∪ (𝑅 ↾ ran 𝑅)))
9	8	eqeq2d 2773	. . . . . . . 8 ⊢ ((𝑅 ↾ dom 𝑅) = 𝑅 → ((𝑅 ↾ ∪ ∪ 𝑅) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) ↔ (𝑅 ↾ ∪ ∪ 𝑅) = (𝑅 ∪ (𝑅 ↾ ran 𝑅))))
10		eqtr 2782	. . . . . . . . 9 ⊢ (((𝑅 ↾ ∪ ∪ 𝑅) = (𝑅 ∪ (𝑅 ↾ ran 𝑅)) ∧ (𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅) → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅)
11	10	ex 417	. . . . . . . 8 ⊢ ((𝑅 ↾ ∪ ∪ 𝑅) = (𝑅 ∪ (𝑅 ↾ ran 𝑅)) → ((𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅 → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅))
12	9, 11	biimtrdi 256	. . . . . . 7 ⊢ ((𝑅 ↾ dom 𝑅) = 𝑅 → ((𝑅 ↾ ∪ ∪ 𝑅) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) → ((𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅 → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅)))
13	12	com3r 88	. . . . . 6 ⊢ ((𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅 → ((𝑅 ↾ dom 𝑅) = 𝑅 → ((𝑅 ↾ ∪ ∪ 𝑅) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅)))
14	7, 13	sylbi 220	. . . . 5 ⊢ ((𝑅 ↾ ran 𝑅) ⊆ 𝑅 → ((𝑅 ↾ dom 𝑅) = 𝑅 → ((𝑅 ↾ ∪ ∪ 𝑅) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅)))
15	5, 6, 14	mpsyl 69	. . . 4 ⊢ (Rel 𝑅 → ((𝑅 ↾ ∪ ∪ 𝑅) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅))
16	4, 15	syl5com 32	. . 3 ⊢ (((𝑅 ↾ ∪ ∪ 𝑅) = (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) ∧ (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅))) → (Rel 𝑅 → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅))
17	2, 3, 16	sylancl 597	. 2 ⊢ (Rel 𝑅 → (Rel 𝑅 → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅))
18	17	pm2.43i 53	1 ⊢ (Rel 𝑅 → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∪ cun 3902   ⊆ wss 3904  ∪ cuni 4871  dom cdm 5660  ran crn 5661   ↾ cres 5662  Rel wrel 5665

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-xp 5666  df-rel 5667  df-cnv 5668  df-dm 5670  df-rn 5671  df-res 5672

This theorem is used by:  relcoi1  6279