Theorem relresfld 6293

Description: Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.) (Proof shortened by Eric Schmidt, 16-Aug-2026.)

Assertion

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

Proof of Theorem relresfld

Step	Hyp	Ref	Expression
1		relfld 6292	. . 3 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))
2	1	reseq2d 5980	. 2 ⊢ (Rel 𝑅 → (𝑅 ↾ ∪ ∪ 𝑅) = (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)))
3		ssun1 4179	. . 3 ⊢ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
4		relssres 6022	. . 3 ⊢ ((Rel 𝑅 ∧ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)) → (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) = 𝑅)
5	3, 4	mpan2 704	. 2 ⊢ (Rel 𝑅 → (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) = 𝑅)
6	2, 5	eqtrd 2798	1 ⊢ (Rel 𝑅 → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ⊆ wss 3900   ∪ cun 4152  ∪ cuni 4870  dom cdm 5655  ran crn 5656   ↾ cres 5657  Rel wrel 5660

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667

This theorem is used by:  relcoi1  6296