Theorem reliun 5806

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 5013	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ (V × V) ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ (V × V))
2		df-rel 5671	. 2 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ (V × V))
3		df-rel 5671	. . 3 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V))
4	3	ralbii 3117	. 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)

Syntax hints:   ↔ wb 209  ∀wral 3085  Vcvv 3463   ⊆ wss 3913  ∪ ciun 4960   × cxp 5662  Rel wrel 5669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-11 2198  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-v 3465  df-ss 3930  df-iun 4962  df-rel 5671

This theorem is referenced by:  reluni  5808  eliunxp  5826  opeliunxp2  5827  dfco2  6249  coiun  6261  fvn0ssdmfun  7072  opeliunxp2f  8208  fsumcom2  15827  fprodcom2  16040  imasaddfnlem  17584  imasvscafn  17593  gsum2d2lem  20045  gsum2d2  20046  gsumcom2  20047  dprd2d2  20118  cnextrel  24191  reldv  26000  dfcnv2  32963  gsumpart  33326  gsumwrd2dccat  33341  cvmliftlem1  35712  cnviun  44305  coiun1  44307  eliunxp2  49036