Theorem relresfld 6267

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 6266	. . 3 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))
2	1	reseq2d 5966	. 2 ⊢ (Rel 𝑅 → (𝑅 ↾ ∪ ∪ 𝑅) = (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)))
3		ssun1 4123	. . 3 ⊢ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
4		relssres 6009	. . 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 2795	1 ⊢ (Rel 𝑅 → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∪ cun 3896   ⊆ wss 3898  ∪ cuni 4866  dom cdm 5647  ran crn 5648   ↾ cres 5649  Rel wrel 5652

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 2732  ax-sep 5248  ax-pr 5390

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 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-xp 5653  df-rel 5654  df-cnv 5655  df-dm 5657  df-rn 5658  df-res 5659

This theorem is used by:  relcoi1  6270