Theorem reliin 5727

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	⊢ (∃𝑥 ∈ 𝐴 Rel 𝐵 → Rel ∩ 𝑥 ∈ 𝐴 𝐵)

Proof of Theorem reliin

Step	Hyp	Ref	Expression
1		iinss 4986	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ⊆ (V × V) → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ (V × V))
2		df-rel 5596	. . 3 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V))
3	2	rexbii 3181	. 2 ⊢ (∃𝑥 ∈ 𝐴 Rel 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝐵 ⊆ (V × V))
4		df-rel 5596	. 2 ⊢ (Rel ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ (V × V))
5	1, 3, 4	3imtr4i 292	1 ⊢ (∃𝑥 ∈ 𝐴 Rel 𝐵 → Rel ∩ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4  ∃wrex 3065  Vcvv 3432   ⊆ wss 3887  ∩ ciin 4925   × cxp 5587  Rel wrel 5594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-v 3434  df-in 3894  df-ss 3904  df-iin 4927  df-rel 5596

This theorem is referenced by:  relint  5729  xpiindi  5744  dibglbN  39180  dihglbcpreN  39314