Theorem ssiinf 4016

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		vex 2863	. . . . 5 ⊢ y ∈ V
2		eliin 3975	. . . . 5 ⊢ (y ∈ V → (y ∈ ∩x ∈ A B ↔ ∀x ∈ A y ∈ B))
3	1, 2	ax-mp 5	. . . 4 ⊢ (y ∈ ∩x ∈ A B ↔ ∀x ∈ A y ∈ B)
4	3	ralbii 2639	. . 3 ⊢ (∀y ∈ C y ∈ ∩x ∈ A B ↔ ∀y ∈ C ∀x ∈ A y ∈ B)
5		ssiinf.1	. . . 4 ⊢ ℲxC
6		nfcv 2490	. . . 4 ⊢ ℲyA
7	5, 6	ralcomf 2770	. . 3 ⊢ (∀y ∈ C ∀x ∈ A y ∈ B ↔ ∀x ∈ A ∀y ∈ C y ∈ B)
8	4, 7	bitri 240	. 2 ⊢ (∀y ∈ C y ∈ ∩x ∈ A B ↔ ∀x ∈ A ∀y ∈ C y ∈ B)
9		dfss3 3264	. 2 ⊢ (C ⊆ ∩x ∈ A B ↔ ∀y ∈ C y ∈ ∩x ∈ A B)
10		dfss3 3264	. . 3 ⊢ (C ⊆ B ↔ ∀y ∈ C y ∈ B)
11	10	ralbii 2639	. 2 ⊢ (∀x ∈ A C ⊆ B ↔ ∀x ∈ A ∀y ∈ C y ∈ B)
12	8, 9, 11	3bitr4i 268	1 ⊢ (C ⊆ ∩x ∈ A B ↔ ∀x ∈ A C ⊆ B)

This proof depends on syntax axioms:   ↔ wb 176   ∈ wcel 1710  Ⅎwnfc 2477  ∀wral 2615  Vcvv 2860   ⊆ wss 3258  ∩ciin 3971

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-iin 3973

This theorem is used by:  ssiin  4017  dmiin  4966