Theorem relresfld 6265

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 6264	. . . 4 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))
2	1	reseq2d 5967	. . 3 ⊢ (Rel 𝑅 → (𝑅 ↾ ∪ ∪ 𝑅) = (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)))
3		resundi 5981	. . 3 ⊢ (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅))
4		eqtr 2784	. . . 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 5989	. . . . 5 ⊢ (𝑅 ↾ ran 𝑅) ⊆ 𝑅
6		resdm 6014	. . . . 5 ⊢ (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
7		ssequn2 4143	. . . . . 6 ⊢ ((𝑅 ↾ ran 𝑅) ⊆ 𝑅 ↔ (𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅)
8		uneq1 4116	. . . . . . . . 9 ⊢ ((𝑅 ↾ dom 𝑅) = 𝑅 → ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) = (𝑅 ∪ (𝑅 ↾ ran 𝑅)))
9	8	eqeq2d 2775	. . . . . . . 8 ⊢ ((𝑅 ↾ dom 𝑅) = 𝑅 → ((𝑅 ↾ ∪ ∪ 𝑅) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) ↔ (𝑅 ↾ ∪ ∪ 𝑅) = (𝑅 ∪ (𝑅 ↾ ran 𝑅))))
10		eqtr 2784	. . . . . . . . 9 ⊢ (((𝑅 ↾ ∪ ∪ 𝑅) = (𝑅 ∪ (𝑅 ↾ ran 𝑅)) ∧ (𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅) → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅)
11	10	ex 416	. . . . . . . 8 ⊢ ((𝑅 ↾ ∪ ∪ 𝑅) = (𝑅 ∪ (𝑅 ↾ ran 𝑅)) → ((𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅 → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅))
12	9, 11	biimtrdi 255	. . . . . . 7 ⊢ ((𝑅 ↾ dom 𝑅) = 𝑅 → ((𝑅 ↾ ∪ ∪ 𝑅) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) → ((𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅 → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅)))
13	12	com3r 87	. . . . . 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 219	. . . . 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 68	. . . 4 ⊢ (Rel 𝑅 → ((𝑅 ↾ ∪ ∪ 𝑅) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅))
16	4, 15	syl5com 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))
17	2, 3, 16	sylancl 595	. 2 ⊢ (Rel 𝑅 → (Rel 𝑅 → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅))
18	17	pm2.43i 52	1 ⊢ (Rel 𝑅 → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1562   ∪ cun 3904   ⊆ wss 3906  ∪ cuni 4867  dom cdm 5649  ran crn 5650   ↾ cres 5651  Rel wrel 5654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-xp 5655  df-rel 5656  df-cnv 5657  df-dm 5659  df-rn 5660  df-res 5661

This theorem is referenced by:  relcoi1  6267