Theorem relresfld 6126

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 6125	. . . 4 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))
2	1	reseq2d 5852	. . 3 ⊢ (Rel 𝑅 → (𝑅 ↾ ∪ ∪ 𝑅) = (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)))
3		resundi 5866	. . 3 ⊢ (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅))
4		eqtr 2841	. . . 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 5877	. . . . 5 ⊢ (𝑅 ↾ ran 𝑅) ⊆ 𝑅
6		resdm 5896	. . . . 5 ⊢ (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
7		ssequn2 4158	. . . . . 6 ⊢ ((𝑅 ↾ ran 𝑅) ⊆ 𝑅 ↔ (𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅)
8		uneq1 4131	. . . . . . . . 9 ⊢ ((𝑅 ↾ dom 𝑅) = 𝑅 → ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) = (𝑅 ∪ (𝑅 ↾ ran 𝑅)))
9	8	eqeq2d 2832	. . . . . . . 8 ⊢ ((𝑅 ↾ dom 𝑅) = 𝑅 → ((𝑅 ↾ ∪ ∪ 𝑅) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) ↔ (𝑅 ↾ ∪ ∪ 𝑅) = (𝑅 ∪ (𝑅 ↾ ran 𝑅))))
10		eqtr 2841	. . . . . . . . 9 ⊢ (((𝑅 ↾ ∪ ∪ 𝑅) = (𝑅 ∪ (𝑅 ↾ ran 𝑅)) ∧ (𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅) → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅)
11	10	ex 415	. . . . . . . 8 ⊢ ((𝑅 ↾ ∪ ∪ 𝑅) = (𝑅 ∪ (𝑅 ↾ ran 𝑅)) → ((𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅 → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅))
12	9, 11	syl6bi 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 588	. 2 ⊢ (Rel 𝑅 → (Rel 𝑅 → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅))
18	17	pm2.43i 52	1 ⊢ (Rel 𝑅 → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∪ cun 3933   ⊆ wss 3935  ∪ cuni 4837  dom cdm 5554  ran crn 5555   ↾ cres 5556  Rel wrel 5559

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-xp 5560  df-rel 5561  df-cnv 5562  df-dm 5564  df-rn 5565  df-res 5566

This theorem is referenced by:  relcoi1  6128