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Theorem reliun 5829
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 4998 . . 3 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}
21releqi 5790 . 2 (Rel 𝑥𝐴 𝐵 ↔ Rel {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵})
3 df-rel 5696 . 2 (Rel {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ⊆ (V × V))
4 abss 4073 . . 3 ({𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ⊆ (V × V) ↔ ∀𝑦(∃𝑥𝐴 𝑦𝐵𝑦 ∈ (V × V)))
5 df-rel 5696 . . . . . 6 (Rel 𝐵𝐵 ⊆ (V × V))
6 df-ss 3980 . . . . . 6 (𝐵 ⊆ (V × V) ↔ ∀𝑦(𝑦𝐵𝑦 ∈ (V × V)))
75, 6bitri 275 . . . . 5 (Rel 𝐵 ↔ ∀𝑦(𝑦𝐵𝑦 ∈ (V × V)))
87ralbii 3091 . . . 4 (∀𝑥𝐴 Rel 𝐵 ↔ ∀𝑥𝐴𝑦(𝑦𝐵𝑦 ∈ (V × V)))
9 ralcom4 3284 . . . 4 (∀𝑥𝐴𝑦(𝑦𝐵𝑦 ∈ (V × V)) ↔ ∀𝑦𝑥𝐴 (𝑦𝐵𝑦 ∈ (V × V)))
10 r19.23v 3181 . . . . 5 (∀𝑥𝐴 (𝑦𝐵𝑦 ∈ (V × V)) ↔ (∃𝑥𝐴 𝑦𝐵𝑦 ∈ (V × V)))
1110albii 1816 . . . 4 (∀𝑦𝑥𝐴 (𝑦𝐵𝑦 ∈ (V × V)) ↔ ∀𝑦(∃𝑥𝐴 𝑦𝐵𝑦 ∈ (V × V)))
128, 9, 113bitri 297 . . 3 (∀𝑥𝐴 Rel 𝐵 ↔ ∀𝑦(∃𝑥𝐴 𝑦𝐵𝑦 ∈ (V × V)))
134, 12bitr4i 278 . 2 ({𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ⊆ (V × V) ↔ ∀𝑥𝐴 Rel 𝐵)
142, 3, 133bitri 297 1 (Rel 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 Rel 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1535  wcel 2106  {cab 2712  wral 3059  wrex 3068  Vcvv 3478  wss 3963   ciun 4996   × cxp 5687  Rel wrel 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-ss 3980  df-iun 4998  df-rel 5696
This theorem is referenced by:  reluni  5831  eliunxp  5851  opeliunxp2  5852  dfco2  6267  coiun  6278  fvn0ssdmfun  7094  opeliunxp2f  8234  fsumcom2  15807  fprodcom2  16017  imasaddfnlem  17575  imasvscafn  17584  gsum2d2lem  20006  gsum2d2  20007  gsumcom2  20008  dprd2d2  20079  cnextrel  24087  reldv  25920  dfcnv2  32693  gsumpart  33043  gsumwrd2dccat  33053  cvmliftlem1  35270  cnviun  43640  coiun1  43642  eliunxp2  48179
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