Theorem reliin 4795

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 3978	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ⊆ (V × V) → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ (V × V))
2		df-rel 4680	. . 3 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V))
3	2	rexbii 2512	. 2 ⊢ (∃𝑥 ∈ 𝐴 Rel 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝐵 ⊆ (V × V))
4		df-rel 4680	. 2 ⊢ (Rel ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ (V × V))
5	1, 3, 4	3imtr4i 201	1 ⊢ (∃𝑥 ∈ 𝐴 Rel 𝐵 → Rel ∩ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4  ∃wrex 2484  Vcvv 2771   ⊆ wss 3165  ∩ ciin 3927   × cxp 4671  Rel wrel 4678

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-in 3171  df-ss 3178  df-iin 3929  df-rel 4680

This theorem is referenced by:  relint  4797  xpiindim  4813