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Theorem reliin 5778
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 5021 . 2 (∃𝑥𝐴 𝐵 ⊆ (V × V) → 𝑥𝐴 𝐵 ⊆ (V × V))
2 df-rel 5645 . . 3 (Rel 𝐵𝐵 ⊆ (V × V))
32rexbii 3098 . 2 (∃𝑥𝐴 Rel 𝐵 ↔ ∃𝑥𝐴 𝐵 ⊆ (V × V))
4 df-rel 5645 . 2 (Rel 𝑥𝐴 𝐵 𝑥𝐴 𝐵 ⊆ (V × V))
51, 3, 43imtr4i 292 1 (∃𝑥𝐴 Rel 𝐵 → Rel 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wrex 3074  Vcvv 3448  wss 3915   ciin 4960   × cxp 5636  Rel wrel 5643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-v 3450  df-in 3922  df-ss 3932  df-iin 4962  df-rel 5645
This theorem is referenced by:  relint  5780  xpiindi  5796  dibglbN  39658  dihglbcpreN  39792
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