Theorem relresfldOLD 6272

Description: Obsolete version of relresfld 6271 as of 16-Aug-2026. (Contributed by FL, 15-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
relresfldOLD	⊢ (Rel 𝑅 → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅)

Proof of Theorem relresfldOLD

Step	Hyp	Ref	Expression
1		relfld 6270	. . . 4 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))
2	1	reseq2d 5970	. . 3 ⊢ (Rel 𝑅 → (𝑅 ↾ ∪ ∪ 𝑅) = (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)))
3		resundi 5984	. . 3 ⊢ (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅))
4		eqtr 2781	. . . 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 5992	. . . . 5 ⊢ (𝑅 ↾ ran 𝑅) ⊆ 𝑅
6		resdm 6017	. . . . 5 ⊢ (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
7		ssequn2 4135	. . . . . 6 ⊢ ((𝑅 ↾ ran 𝑅) ⊆ 𝑅 ↔ (𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅)
8		uneq1 4108	. . . . . . . . 9 ⊢ ((𝑅 ↾ dom 𝑅) = 𝑅 → ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) = (𝑅 ∪ (𝑅 ↾ ran 𝑅)))
9	8	eqeq2d 2772	. . . . . . . 8 ⊢ ((𝑅 ↾ dom 𝑅) = 𝑅 → ((𝑅 ↾ ∪ ∪ 𝑅) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) ↔ (𝑅 ↾ ∪ ∪ 𝑅) = (𝑅 ∪ (𝑅 ↾ ran 𝑅))))
10		eqtr 2781	. . . . . . . . 9 ⊢ (((𝑅 ↾ ∪ ∪ 𝑅) = (𝑅 ∪ (𝑅 ↾ ran 𝑅)) ∧ (𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅) → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅)
11	10	ex 418	. . . . . . . 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 598	. 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 401   = wceq 1570   ∪ cun 3897   ⊆ wss 3899  ∪ cuni 4867  dom cdm 5651  ran crn 5652   ↾ cres 5653  Rel wrel 5656

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5657  df-rel 5658  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663

This theorem is used by: (None)