Theorem reliin 5654

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 4943	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ⊆ (V × V) → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ (V × V))
2		df-rel 5526	. . 3 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V))
3	2	rexbii 3210	. 2 ⊢ (∃𝑥 ∈ 𝐴 Rel 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝐵 ⊆ (V × V))
4		df-rel 5526	. 2 ⊢ (Rel ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ (V × V))
5	1, 3, 4	3imtr4i 295	1 ⊢ (∃𝑥 ∈ 𝐴 Rel 𝐵 → Rel ∩ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4  ∃wrex 3107  Vcvv 3441   ⊆ wss 3881  ∩ ciin 4882   × cxp 5517  Rel wrel 5524

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-in 3888  df-ss 3898  df-iin 4884  df-rel 5526

This theorem is referenced by:  relint  5656  xpiindi  5670  dibglbN  38462  dihglbcpreN  38596