Theorem ssiinf 5018

Description: Subset theorem for an indexed intersection. (Contributed by FL, 15-Oct-2012.) (Proof shortened by Mario Carneiro, 14-Oct-2016.)

Hypothesis

Ref	Expression
ssiinf.1	⊢ Ⅎ𝑥𝐶

Assertion

Ref	Expression
ssiinf	⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)

Proof of Theorem ssiinf

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliin 4961	. . . . 5 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
2	1	elv 3459	. . . 4 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
3	2	ralbii 3112	. . 3 ⊢ (∀𝑦 ∈ 𝐶 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦 ∈ 𝐶 ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
4		ssiinf.1	. . . 4 ⊢ Ⅎ𝑥𝐶
5		nfcv 2926	. . . 4 ⊢ Ⅎ𝑦𝐴
6	4, 5	ralcomf 3304	. . 3 ⊢ (∀𝑦 ∈ 𝐶 ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑦 ∈ 𝐵)
7	3, 6	bitri 278	. 2 ⊢ (∀𝑦 ∈ 𝐶 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑦 ∈ 𝐵)
8		dfss3 3902	. 2 ⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦 ∈ 𝐶 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵)
9		dfss3 3902	. . 3 ⊢ (𝐶 ⊆ 𝐵 ↔ ∀𝑦 ∈ 𝐶 𝑦 ∈ 𝐵)
10	9	ralbii 3112	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑦 ∈ 𝐵)
11	7, 8, 10	3bitr4i 306	1 ⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)

This proof depends on syntax axioms:   ↔ wb 209   ∈ wcel 2147  Ⅎwnfc 2911  ∀wral 3080  Vcvv 3454   ⊆ wss 3900  ∩ ciin 4957

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-v 3456  df-ss 3901  df-iin 4959

This theorem is used by:  ssiin  5019  dmiin  5947