Theorem ssiinf 4020

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 2805	. . . . 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 2538	. . 3 ⊢ (∀𝑦 ∈ 𝐶 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦 ∈ 𝐶 ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
5		ssiinf.1	. . . 4 ⊢ Ⅎ𝑥𝐶
6		nfcv 2374	. . . 4 ⊢ Ⅎ𝑦𝐴
7	5, 6	ralcomf 2694	. . 3 ⊢ (∀𝑦 ∈ 𝐶 ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑦 ∈ 𝐵)
8	4, 7	bitri 184	. 2 ⊢ (∀𝑦 ∈ 𝐶 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑦 ∈ 𝐵)
9		dfss3 3216	. 2 ⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦 ∈ 𝐶 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵)
10		dfss3 3216	. . 3 ⊢ (𝐶 ⊆ 𝐵 ↔ ∀𝑦 ∈ 𝐶 𝑦 ∈ 𝐵)
11	10	ralbii 2538	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑦 ∈ 𝐵)
12	8, 9, 11	3bitr4i 212	1 ⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)

Syntax hints:   ↔ wb 105   ∈ wcel 2202  Ⅎwnfc 2361  ∀wral 2510  Vcvv 2802   ⊆ wss 3200  ∩ ciin 3971

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-in 3206  df-ss 3213  df-iin 3973

This theorem is referenced by:  ssiin  4021  dmiin  4978