Theorem reluni 5761

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 4988	. . 3 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥
2	1	releqi 5721	. 2 ⊢ (Rel ∪ 𝐴 ↔ Rel ∪ 𝑥 ∈ 𝐴 𝑥)
3		reliun 5759	. 2 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 𝑥 ↔ ∀𝑥 ∈ 𝐴 Rel 𝑥)
4	2, 3	bitri 276	1 ⊢ (Rel ∪ 𝐴 ↔ ∀𝑥 ∈ 𝐴 Rel 𝑥)

Syntax hints:   ↔ wb 207  ∀wral 3053  ∪ cuni 4838  ∪ ciun 4921  Rel wrel 5623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-11 2168  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-v 3433  df-ss 3900  df-uni 4839  df-iun 4923  df-rel 5625

This theorem is referenced by:  fununi  6560  frrlem6  8231  tfrlem6  8311  bnj1379  35012