Theorem ssiinf 5019

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 3460	. . . 4 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
3	2	ralbii 3111	. . 3 ⊢ (∀𝑦 ∈ 𝐶 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦 ∈ 𝐶 ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
4		ssiinf.1	. . . 4 ⊢ Ⅎ𝑥𝐶
5		nfcv 2925	. . . 4 ⊢ Ⅎ𝑦𝐴
6	4, 5	ralcomf 3303	. . 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 3926	. 2 ⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦 ∈ 𝐶 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵)
9		dfss3 3926	. . 3 ⊢ (𝐶 ⊆ 𝐵 ↔ ∀𝑦 ∈ 𝐶 𝑦 ∈ 𝐵)
10	9	ralbii 3111	. 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)

Syntax hints:   ↔ wb 209   ∈ wcel 2143  Ⅎwnfc 2910  ∀wral 3079  Vcvv 3455   ⊆ wss 3905  ∩ ciin 4957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-v 3457  df-ss 3922  df-iin 4959

This theorem is referenced by:  ssiin  5020  dmiin  5943