Theorem relint 4671

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 4669	. 2 ⊢ (∃𝑥 ∈ 𝐴 Rel 𝑥 → Rel ∩ 𝑥 ∈ 𝐴 𝑥)
2		intiin 3875	. . 3 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥
3	2	releqi 4630	. 2 ⊢ (Rel ∩ 𝐴 ↔ Rel ∩ 𝑥 ∈ 𝐴 𝑥)
4	1, 3	sylibr 133	1 ⊢ (∃𝑥 ∈ 𝐴 Rel 𝑥 → Rel ∩ 𝐴)

Syntax hints:   → wi 4  ∃wrex 2418  ∩ cint 3779  ∩ ciin 3822  Rel wrel 4552

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-in 3082  df-ss 3089  df-int 3780  df-iin 3824  df-rel 4554

This theorem is referenced by: (None)