Theorem relint 4658

Description: The intersection of a class is a relation if at least one member is a relation. (Contributed by NM, 8-Mar-2014.)

Assertion

Ref	Expression
relint	⊢ (∃𝑥 ∈ 𝐴 Rel 𝑥 → Rel ∩ 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem relint

Step	Hyp	Ref	Expression
1		reliin 4656	. 2 ⊢ (∃𝑥 ∈ 𝐴 Rel 𝑥 → Rel ∩ 𝑥 ∈ 𝐴 𝑥)
2		intiin 3862	. . 3 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥
3	2	releqi 4617	. 2 ⊢ (Rel ∩ 𝐴 ↔ Rel ∩ 𝑥 ∈ 𝐴 𝑥)
4	1, 3	sylibr 133	1 ⊢ (∃𝑥 ∈ 𝐴 Rel 𝑥 → Rel ∩ 𝐴)

Syntax hints:   → wi 4  ∃wrex 2415  ∩ cint 3766  ∩ ciin 3809  Rel wrel 4539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-in 3072  df-ss 3079  df-int 3767  df-iin 3811  df-rel 4541

This theorem is referenced by: (None)