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Theorem reliun 5162
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 4457 . . 3 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}
21releqi 5125 . 2 (Rel 𝑥𝐴 𝐵 ↔ Rel {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵})
3 df-rel 5045 . 2 (Rel {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ⊆ (V × V))
4 abss 3634 . . 3 ({𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ⊆ (V × V) ↔ ∀𝑦(∃𝑥𝐴 𝑦𝐵𝑦 ∈ (V × V)))
5 df-rel 5045 . . . . . 6 (Rel 𝐵𝐵 ⊆ (V × V))
6 dfss2 3557 . . . . . 6 (𝐵 ⊆ (V × V) ↔ ∀𝑦(𝑦𝐵𝑦 ∈ (V × V)))
75, 6bitri 263 . . . . 5 (Rel 𝐵 ↔ ∀𝑦(𝑦𝐵𝑦 ∈ (V × V)))
87ralbii 2963 . . . 4 (∀𝑥𝐴 Rel 𝐵 ↔ ∀𝑥𝐴𝑦(𝑦𝐵𝑦 ∈ (V × V)))
9 ralcom4 3197 . . . 4 (∀𝑥𝐴𝑦(𝑦𝐵𝑦 ∈ (V × V)) ↔ ∀𝑦𝑥𝐴 (𝑦𝐵𝑦 ∈ (V × V)))
10 r19.23v 3005 . . . . 5 (∀𝑥𝐴 (𝑦𝐵𝑦 ∈ (V × V)) ↔ (∃𝑥𝐴 𝑦𝐵𝑦 ∈ (V × V)))
1110albii 1737 . . . 4 (∀𝑦𝑥𝐴 (𝑦𝐵𝑦 ∈ (V × V)) ↔ ∀𝑦(∃𝑥𝐴 𝑦𝐵𝑦 ∈ (V × V)))
128, 9, 113bitri 285 . . 3 (∀𝑥𝐴 Rel 𝐵 ↔ ∀𝑦(∃𝑥𝐴 𝑦𝐵𝑦 ∈ (V × V)))
134, 12bitr4i 266 . 2 ({𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ⊆ (V × V) ↔ ∀𝑥𝐴 Rel 𝐵)
142, 3, 133bitri 285 1 (Rel 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 Rel 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wal 1473  wcel 1977  {cab 2596  wral 2896  wrex 2897  Vcvv 3173  wss 3540   ciun 4455   × cxp 5036  Rel wrel 5043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-in 3547  df-ss 3554  df-iun 4457  df-rel 5045
This theorem is referenced by:  reluni  5164  eliunxp  5181  opeliunxp2  5182  dfco2  5551  coiun  5562  fvn0ssdmfun  6258  opeliunxp2f  7223  fsumcom2  14347  fsumcom2OLD  14348  fprodcom2  14553  fprodcom2OLD  14554  imasaddfnlem  16011  imasvscafn  16020  gsum2d2lem  18195  gsum2d2  18196  gsumcom2  18197  dprd2d2  18266  cnextrel  21677  reldv  23440  dfcnv2  28859  cvmliftlem1  30521  cnviun  36961  coiun1  36963  eliunxp2  41905
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