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Theorem reliin 5677
 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 4966 . 2 (∃𝑥𝐴 𝐵 ⊆ (V × V) → 𝑥𝐴 𝐵 ⊆ (V × V))
2 df-rel 5549 . . 3 (Rel 𝐵𝐵 ⊆ (V × V))
32rexbii 3242 . 2 (∃𝑥𝐴 Rel 𝐵 ↔ ∃𝑥𝐴 𝐵 ⊆ (V × V))
4 df-rel 5549 . 2 (Rel 𝑥𝐴 𝐵 𝑥𝐴 𝐵 ⊆ (V × V))
51, 3, 43imtr4i 295 1 (∃𝑥𝐴 Rel 𝐵 → Rel 𝑥𝐴 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∃wrex 3134  Vcvv 3480   ⊆ wss 3919  ∩ ciin 4906   × cxp 5540  Rel wrel 5547 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-in 3926  df-ss 3936  df-iin 4908  df-rel 5549 This theorem is referenced by:  relint  5679  xpiindi  5693  dibglbN  38372  dihglbcpreN  38506
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