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Theorem reliun 5410
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 4680 . . 3 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}
21releqi 5374 . 2 (Rel 𝑥𝐴 𝐵 ↔ Rel {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵})
3 df-rel 5286 . 2 (Rel {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ⊆ (V × V))
4 abss 3833 . . 3 ({𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ⊆ (V × V) ↔ ∀𝑦(∃𝑥𝐴 𝑦𝐵𝑦 ∈ (V × V)))
5 df-rel 5286 . . . . . 6 (Rel 𝐵𝐵 ⊆ (V × V))
6 dfss2 3751 . . . . . 6 (𝐵 ⊆ (V × V) ↔ ∀𝑦(𝑦𝐵𝑦 ∈ (V × V)))
75, 6bitri 266 . . . . 5 (Rel 𝐵 ↔ ∀𝑦(𝑦𝐵𝑦 ∈ (V × V)))
87ralbii 3127 . . . 4 (∀𝑥𝐴 Rel 𝐵 ↔ ∀𝑥𝐴𝑦(𝑦𝐵𝑦 ∈ (V × V)))
9 ralcom4 3377 . . . 4 (∀𝑥𝐴𝑦(𝑦𝐵𝑦 ∈ (V × V)) ↔ ∀𝑦𝑥𝐴 (𝑦𝐵𝑦 ∈ (V × V)))
10 r19.23v 3170 . . . . 5 (∀𝑥𝐴 (𝑦𝐵𝑦 ∈ (V × V)) ↔ (∃𝑥𝐴 𝑦𝐵𝑦 ∈ (V × V)))
1110albii 1914 . . . 4 (∀𝑦𝑥𝐴 (𝑦𝐵𝑦 ∈ (V × V)) ↔ ∀𝑦(∃𝑥𝐴 𝑦𝐵𝑦 ∈ (V × V)))
128, 9, 113bitri 288 . . 3 (∀𝑥𝐴 Rel 𝐵 ↔ ∀𝑦(∃𝑥𝐴 𝑦𝐵𝑦 ∈ (V × V)))
134, 12bitr4i 269 . 2 ({𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ⊆ (V × V) ↔ ∀𝑥𝐴 Rel 𝐵)
142, 3, 133bitri 288 1 (Rel 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 Rel 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wal 1650  wcel 2155  {cab 2751  wral 3055  wrex 3056  Vcvv 3350  wss 3734   ciun 4678   × cxp 5277  Rel wrel 5284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-v 3352  df-in 3741  df-ss 3748  df-iun 4680  df-rel 5286
This theorem is referenced by:  reluni  5412  eliunxp  5430  opeliunxp2  5431  dfco2  5822  coiun  5833  fvn0ssdmfun  6542  opeliunxp2f  7541  fsumcom2  14793  fprodcom2  15000  imasaddfnlem  16457  imasvscafn  16466  gsum2d2lem  18641  gsum2d2  18642  gsumcom2  18643  dprd2d2  18713  cnextrel  22149  reldv  23928  dfcnv2  29928  cvmliftlem1  31718  cnviun  38620  coiun1  38622  eliunxp2  42784
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