Theorem reliun 5766

Description: An indexed union is a relation iff each member of its indexed family is a relation. (Contributed by NM, 19-Dec-2008.) (Proof shortened by SN, 2-Feb-2025.)

Assertion

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

Proof of Theorem reliun

Step	Hyp	Ref	Expression
1		iunss 5001	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ (V × V) ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ (V × V))
2		df-rel 5632	. 2 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ (V × V))
3		df-rel 5632	. . 3 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V))
4	3	ralbii 3083	. 2 ⊢ (∀𝑥 ∈ 𝐴 Rel 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ (V × V))
5	1, 2, 4	3bitr4i 303	1 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 Rel 𝐵)

Syntax hints:   ↔ wb 206  ∀wral 3052  Vcvv 3441   ⊆ wss 3902  ∪ ciun 4947   × cxp 5623  Rel wrel 5630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3062  df-v 3443  df-ss 3919  df-iun 4949  df-rel 5632

This theorem is referenced by:  reluni  5768  eliunxp  5787  opeliunxp2  5788  dfco2  6204  coiun  6216  fvn0ssdmfun  7021  opeliunxp2f  8154  fsumcom2  15701  fprodcom2  15911  imasaddfnlem  17453  imasvscafn  17462  gsum2d2lem  19906  gsum2d2  19907  gsumcom2  19908  dprd2d2  19979  cnextrel  24011  reldv  25831  dfcnv2  32756  gsumpart  33148  gsumwrd2dccat  33162  cvmliftlem1  35481  cnviun  43958  coiun1  43960  eliunxp2  48647