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Theorem reliun 5816
Description: An indexed union is a relation iff each member of its indexed family is a relation. (Contributed by NM, 19-Dec-2008.)
Assertion
Ref Expression
reliun (Rel 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 Rel 𝐵)

Proof of Theorem reliun
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-iun 4999 . . 3 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}
21releqi 5777 . 2 (Rel 𝑥𝐴 𝐵 ↔ Rel {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵})
3 df-rel 5683 . 2 (Rel {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ⊆ (V × V))
4 abss 4057 . . 3 ({𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ⊆ (V × V) ↔ ∀𝑦(∃𝑥𝐴 𝑦𝐵𝑦 ∈ (V × V)))
5 df-rel 5683 . . . . . 6 (Rel 𝐵𝐵 ⊆ (V × V))
6 dfss2 3968 . . . . . 6 (𝐵 ⊆ (V × V) ↔ ∀𝑦(𝑦𝐵𝑦 ∈ (V × V)))
75, 6bitri 274 . . . . 5 (Rel 𝐵 ↔ ∀𝑦(𝑦𝐵𝑦 ∈ (V × V)))
87ralbii 3093 . . . 4 (∀𝑥𝐴 Rel 𝐵 ↔ ∀𝑥𝐴𝑦(𝑦𝐵𝑦 ∈ (V × V)))
9 ralcom4 3283 . . . 4 (∀𝑥𝐴𝑦(𝑦𝐵𝑦 ∈ (V × V)) ↔ ∀𝑦𝑥𝐴 (𝑦𝐵𝑦 ∈ (V × V)))
10 r19.23v 3182 . . . . 5 (∀𝑥𝐴 (𝑦𝐵𝑦 ∈ (V × V)) ↔ (∃𝑥𝐴 𝑦𝐵𝑦 ∈ (V × V)))
1110albii 1821 . . . 4 (∀𝑦𝑥𝐴 (𝑦𝐵𝑦 ∈ (V × V)) ↔ ∀𝑦(∃𝑥𝐴 𝑦𝐵𝑦 ∈ (V × V)))
128, 9, 113bitri 296 . . 3 (∀𝑥𝐴 Rel 𝐵 ↔ ∀𝑦(∃𝑥𝐴 𝑦𝐵𝑦 ∈ (V × V)))
134, 12bitr4i 277 . 2 ({𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ⊆ (V × V) ↔ ∀𝑥𝐴 Rel 𝐵)
142, 3, 133bitri 296 1 (Rel 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 Rel 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1539  wcel 2106  {cab 2709  wral 3061  wrex 3070  Vcvv 3474  wss 3948   ciun 4997   × cxp 5674  Rel wrel 5681
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-v 3476  df-in 3955  df-ss 3965  df-iun 4999  df-rel 5683
This theorem is referenced by:  reluni  5818  eliunxp  5837  opeliunxp2  5838  dfco2  6244  coiun  6255  fvn0ssdmfun  7076  opeliunxp2f  8194  fsumcom2  15719  fprodcom2  15927  imasaddfnlem  17473  imasvscafn  17482  gsum2d2lem  19840  gsum2d2  19841  gsumcom2  19842  dprd2d2  19913  cnextrel  23566  reldv  25386  dfcnv2  31896  gsumpart  32202  cvmliftlem1  34271  cnviun  42391  coiun1  42393  eliunxp2  46999
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