Theorem reluni 5799

Description: The union of a class is a relation iff any member is a relation. Exercise 6 of [TakeutiZaring] p. 25 and its converse. (Contributed by NM, 13-Aug-2004.)

Assertion

Ref	Expression
reluni	⊢ (Rel ∪ 𝐴 ↔ ∀𝑥 ∈ 𝐴 Rel 𝑥)

Distinct variable group:   𝑥,𝐴

Proof of Theorem reluni

Step	Hyp	Ref	Expression
1		uniiun 5017	. . 3 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥
2	1	releqi 5758	. 2 ⊢ (Rel ∪ 𝐴 ↔ Rel ∪ 𝑥 ∈ 𝐴 𝑥)
3		reliun 5797	. 2 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 𝑥 ↔ ∀𝑥 ∈ 𝐴 Rel 𝑥)
4	2, 3	bitri 278	1 ⊢ (Rel ∪ 𝐴 ↔ ∀𝑥 ∈ 𝐴 Rel 𝑥)

This proof depends on syntax axioms:   ↔ wb 209  ∀wral 3076  ∪ cuni 4867  ∪ ciun 4951  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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-v 3452  df-ss 3916  df-uni 4868  df-iun 4953  df-rel 5662

This theorem is used by:  fununi  6609  frrlem6  8291  tfrlem6OLD  8372  bnj1379  35340