Theorem relint 4851

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 4849	. 2 ⊢ (∃𝑥 ∈ 𝐴 Rel 𝑥 → Rel ∩ 𝑥 ∈ 𝐴 𝑥)
2		intiin 4025	. . 3 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥
3	2	releqi 4809	. 2 ⊢ (Rel ∩ 𝐴 ↔ Rel ∩ 𝑥 ∈ 𝐴 𝑥)
4	1, 3	sylibr 134	1 ⊢ (∃𝑥 ∈ 𝐴 Rel 𝑥 → Rel ∩ 𝐴)

Syntax hints:   → wi 4  ∃wrex 2511  ∩ cint 3928  ∩ ciin 3971  Rel wrel 4730

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-in 3206  df-ss 3213  df-int 3929  df-iin 3973  df-rel 4732

This theorem is referenced by: (None)