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Theorem reliun 4486
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 3687 . . 3 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}
21releqi 4451 . 2 (Rel 𝑥𝐴 𝐵 ↔ Rel {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵})
3 df-rel 4380 . 2 (Rel {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ⊆ (V × V))
4 abss 3037 . . 3 ({𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ⊆ (V × V) ↔ ∀𝑦(∃𝑥𝐴 𝑦𝐵𝑦 ∈ (V × V)))
5 df-rel 4380 . . . . . 6 (Rel 𝐵𝐵 ⊆ (V × V))
6 dfss2 2962 . . . . . 6 (𝐵 ⊆ (V × V) ↔ ∀𝑦(𝑦𝐵𝑦 ∈ (V × V)))
75, 6bitri 177 . . . . 5 (Rel 𝐵 ↔ ∀𝑦(𝑦𝐵𝑦 ∈ (V × V)))
87ralbii 2347 . . . 4 (∀𝑥𝐴 Rel 𝐵 ↔ ∀𝑥𝐴𝑦(𝑦𝐵𝑦 ∈ (V × V)))
9 ralcom4 2593 . . . 4 (∀𝑥𝐴𝑦(𝑦𝐵𝑦 ∈ (V × V)) ↔ ∀𝑦𝑥𝐴 (𝑦𝐵𝑦 ∈ (V × V)))
10 r19.23v 2442 . . . . 5 (∀𝑥𝐴 (𝑦𝐵𝑦 ∈ (V × V)) ↔ (∃𝑥𝐴 𝑦𝐵𝑦 ∈ (V × V)))
1110albii 1375 . . . 4 (∀𝑦𝑥𝐴 (𝑦𝐵𝑦 ∈ (V × V)) ↔ ∀𝑦(∃𝑥𝐴 𝑦𝐵𝑦 ∈ (V × V)))
128, 9, 113bitri 199 . . 3 (∀𝑥𝐴 Rel 𝐵 ↔ ∀𝑦(∃𝑥𝐴 𝑦𝐵𝑦 ∈ (V × V)))
134, 12bitr4i 180 . 2 ({𝑦 ∣ ∃𝑥𝐴 𝑦𝐵} ⊆ (V × V) ↔ ∀𝑥𝐴 Rel 𝐵)
142, 3, 133bitri 199 1 (Rel 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 Rel 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wal 1257  wcel 1409  {cab 2042  wral 2323  wrex 2324  Vcvv 2574  wss 2945   ciun 3685   × cxp 4371  Rel wrel 4378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-in 2952  df-ss 2959  df-iun 3687  df-rel 4380
This theorem is referenced by:  reluni  4488  eliunxp  4503  opeliunxp2  4504  dfco2  4848  coiun  4858
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