Theorem reliun 5799

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 5006	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ (V × V) ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ (V × V))
2		df-rel 5662	. 2 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ (V × V))
3		df-rel 5662	. . 3 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V))
4	3	ralbii 3112	. 2 ⊢ (∀𝑥 ∈ 𝐴 Rel 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ (V × V))
5	1, 2, 4	3bitr4i 306	1 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 Rel 𝐵)

This proof depends on syntax axioms:   ↔ wb 209  ∀wral 3080  Vcvv 3454   ⊆ wss 3900  ∪ ciun 4954   × cxp 5653  Rel wrel 5660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-11 2196  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-v 3456  df-ss 3901  df-iun 4956  df-rel 5662

This theorem is used by:  reluni  5801  eliunxp  5819  opeliunxp2  5820  dfco2  6253  coiun  6265  fvn0ssdmfun  7085  opeliunxp2f  8241  fsumcom2  15955  fprodcom2  16166  imasaddfnlem  17715  imasvscafn  17724  gsum2d2lem  20202  gsum2d2  20203  gsumcom2  20204  dprd2d2  20275  cnextrel  24397  reldv  26205  dfcnv2  33326  gsumpart  33680  gsumwrd2dccat  33695  cvmliftlem1  36174  cnviun  44796  coiun1  44798  eliunxp2  49591