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Theorem reliin 5163
 Description: An indexed intersection is a relation if at least one of the member of the indexed family is a relation. (Contributed by NM, 8-Mar-2014.)
Assertion
Ref Expression
reliin (∃𝑥𝐴 Rel 𝐵 → Rel 𝑥𝐴 𝐵)

Proof of Theorem reliin
StepHypRef Expression
1 iinss 4507 . 2 (∃𝑥𝐴 𝐵 ⊆ (V × V) → 𝑥𝐴 𝐵 ⊆ (V × V))
2 df-rel 5045 . . 3 (Rel 𝐵𝐵 ⊆ (V × V))
32rexbii 3023 . 2 (∃𝑥𝐴 Rel 𝐵 ↔ ∃𝑥𝐴 𝐵 ⊆ (V × V))
4 df-rel 5045 . 2 (Rel 𝑥𝐴 𝐵 𝑥𝐴 𝐵 ⊆ (V × V))
51, 3, 43imtr4i 280 1 (∃𝑥𝐴 Rel 𝐵 → Rel 𝑥𝐴 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ∩ ciin 4456   × 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-iin 4458  df-rel 5045 This theorem is referenced by:  relint  5165  xpiindi  5179  dibglbN  35473  dihglbcpreN  35607
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