Theorem reliin 5760

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 5625	. . 3 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V))
3	2	rexbii 3086	. 2 ⊢ (∃𝑥 ∈ 𝐴 Rel 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝐵 ⊆ (V × V))
4		df-rel 5625	. 2 ⊢ (Rel ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ (V × V))
5	1, 3, 4	3imtr4i 293	1 ⊢ (∃𝑥 ∈ 𝐴 Rel 𝐵 → Rel ∩ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4  ∃wrex 3063  Vcvv 3431   ⊆ wss 3883  ∩ ciin 4922   × cxp 5616  Rel wrel 5623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-v 3433  df-ss 3900  df-iin 4924  df-rel 5625

This theorem is referenced by:  relint  5762  xpiindi  5777  dibglbN  41658  dihglbcpreN  41792  iinxp  49321