Theorem pw1in 4165

Description: Unit power class distributes over intersection. (Contributed by SF, 13-Feb-2015.)

Assertion

Ref	Expression
pw1in	⊢ 𝒫1(A ∩ B) = (𝒫1A ∩ 𝒫1B)

Proof of Theorem pw1in

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ancom 437	. . . . 5 ⊢ (((y ∈ A ∧ x ∈ 𝒫1B) ∧ x = {y}) ↔ (x = {y} ∧ (y ∈ A ∧ x ∈ 𝒫1B)))
2		eleq1 2413	. . . . . . . . 9 ⊢ (x = {y} → (x ∈ 𝒫1B ↔ {y} ∈ 𝒫1B))
3		snelpw1 4147	. . . . . . . . 9 ⊢ ({y} ∈ 𝒫1B ↔ y ∈ B)
4	2, 3	syl6bb 252	. . . . . . . 8 ⊢ (x = {y} → (x ∈ 𝒫1B ↔ y ∈ B))
5	4	anbi2d 684	. . . . . . 7 ⊢ (x = {y} → ((y ∈ A ∧ x ∈ 𝒫1B) ↔ (y ∈ A ∧ y ∈ B)))
6		elin 3220	. . . . . . 7 ⊢ (y ∈ (A ∩ B) ↔ (y ∈ A ∧ y ∈ B))
7	5, 6	syl6bbr 254	. . . . . 6 ⊢ (x = {y} → ((y ∈ A ∧ x ∈ 𝒫1B) ↔ y ∈ (A ∩ B)))
8	7	pm5.32ri 619	. . . . 5 ⊢ (((y ∈ A ∧ x ∈ 𝒫1B) ∧ x = {y}) ↔ (y ∈ (A ∩ B) ∧ x = {y}))
9		an12 772	. . . . 5 ⊢ ((x = {y} ∧ (y ∈ A ∧ x ∈ 𝒫1B)) ↔ (y ∈ A ∧ (x = {y} ∧ x ∈ 𝒫1B)))
10	1, 8, 9	3bitr3i 266	. . . 4 ⊢ ((y ∈ (A ∩ B) ∧ x = {y}) ↔ (y ∈ A ∧ (x = {y} ∧ x ∈ 𝒫1B)))
11	10	rexbii2 2644	. . 3 ⊢ (∃y ∈ (A ∩ B)x = {y} ↔ ∃y ∈ A (x = {y} ∧ x ∈ 𝒫1B))
12		elpw1 4145	. . 3 ⊢ (x ∈ 𝒫1(A ∩ B) ↔ ∃y ∈ (A ∩ B)x = {y})
13		elpw1 4145	. . . . 5 ⊢ (x ∈ 𝒫1A ↔ ∃y ∈ A x = {y})
14	13	anbi1i 676	. . . 4 ⊢ ((x ∈ 𝒫1A ∧ x ∈ 𝒫1B) ↔ (∃y ∈ A x = {y} ∧ x ∈ 𝒫1B))
15		elin 3220	. . . 4 ⊢ (x ∈ (𝒫1A ∩ 𝒫1B) ↔ (x ∈ 𝒫1A ∧ x ∈ 𝒫1B))
16		r19.41v 2765	. . . 4 ⊢ (∃y ∈ A (x = {y} ∧ x ∈ 𝒫1B) ↔ (∃y ∈ A x = {y} ∧ x ∈ 𝒫1B))
17	14, 15, 16	3bitr4i 268	. . 3 ⊢ (x ∈ (𝒫1A ∩ 𝒫1B) ↔ ∃y ∈ A (x = {y} ∧ x ∈ 𝒫1B))
18	11, 12, 17	3bitr4i 268	. 2 ⊢ (x ∈ 𝒫1(A ∩ B) ↔ x ∈ (𝒫1A ∩ 𝒫1B))
19	18	eqriv 2350	1 ⊢ 𝒫1(A ∩ B) = (𝒫1A ∩ 𝒫1B)

This proof depends on syntax axioms:   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2616   ∩ cin 3209  {csn 3738  𝒫₁cpw1 4136

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  ax-nin 4079  ax-sn 4088

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-ne 2519  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-pw 3725  df-sn 3742  df-1c 4137  df-pw1 4138

This theorem is used by:  tfindi  4497  tcdi  6165  ce0addcnnul  6180