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Theorem reliin 4766
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 3953 . 2 (∃𝑥𝐴 𝐵 ⊆ (V × V) → 𝑥𝐴 𝐵 ⊆ (V × V))
2 df-rel 4651 . . 3 (Rel 𝐵𝐵 ⊆ (V × V))
32rexbii 2497 . 2 (∃𝑥𝐴 Rel 𝐵 ↔ ∃𝑥𝐴 𝐵 ⊆ (V × V))
4 df-rel 4651 . 2 (Rel 𝑥𝐴 𝐵 𝑥𝐴 𝐵 ⊆ (V × V))
51, 3, 43imtr4i 201 1 (∃𝑥𝐴 Rel 𝐵 → Rel 𝑥𝐴 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wrex 2469  Vcvv 2752  wss 3144   ciin 3902   × cxp 4642  Rel wrel 4649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-in 3150  df-ss 3157  df-iin 3904  df-rel 4651
This theorem is referenced by:  relint  4768  xpiindim  4782
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