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Theorem reliin 5827
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 5056 . 2 (∃𝑥𝐴 𝐵 ⊆ (V × V) → 𝑥𝐴 𝐵 ⊆ (V × V))
2 df-rel 5692 . . 3 (Rel 𝐵𝐵 ⊆ (V × V))
32rexbii 3094 . 2 (∃𝑥𝐴 Rel 𝐵 ↔ ∃𝑥𝐴 𝐵 ⊆ (V × V))
4 df-rel 5692 . 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 3070  Vcvv 3480  wss 3951   ciin 4992   × cxp 5683  Rel wrel 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-v 3482  df-ss 3968  df-iin 4994  df-rel 5692
This theorem is referenced by:  relint  5829  xpiindi  5846  dibglbN  41168  dihglbcpreN  41302
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