Theorem reliun 5767

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 4988	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ (V × V) ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ (V × V))
2		df-rel 5633	. 2 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ (V × V))
3		df-rel 5633	. . 3 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V))
4	3	ralbii 3084	. 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 3430   ⊆ wss 3890  ∪ ciun 4934   × cxp 5624  Rel wrel 5631

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 3063  df-v 3432  df-ss 3907  df-iun 4936  df-rel 5633

This theorem is referenced by:  reluni  5769  eliunxp  5788  opeliunxp2  5789  dfco2  6205  coiun  6217  fvn0ssdmfun  7022  opeliunxp2f  8155  fsumcom2  15731  fprodcom2  15944  imasaddfnlem  17487  imasvscafn  17496  gsum2d2lem  19943  gsum2d2  19944  gsumcom2  19945  dprd2d2  20016  cnextrel  24042  reldv  25851  dfcnv2  32767  gsumpart  33143  gsumwrd2dccat  33158  cvmliftlem1  35487  cnviun  44099  coiun1  44101  eliunxp2  48826