Theorem pw1equn 4332

Description: A condition for a unit power class to equal a union. (Contributed by SF, 26-Jan-2015.)

Hypotheses

Ref	Expression
pw1equn.1	⊢ A ∈ V
pw1equn.2	⊢ B ∈ V

Assertion

Ref	Expression
pw1equn	⊢ (𝒫1C = (A ∪ B) ↔ ∃x∃y(C = (x ∪ y) ∧ A = 𝒫1x ∧ B = 𝒫1y))

Distinct variable groups:   x,A,y   x,B,y   x,C,y

Proof of Theorem pw1equn

Step	Hyp	Ref	Expression
1		unipw1 4326	. . . 4 ⊢ ∪𝒫1C = C
2		unieq 3901	. . . 4 ⊢ (𝒫1C = (A ∪ B) → ∪𝒫1C = ∪(A ∪ B))
3	1, 2	syl5eqr 2399	. . 3 ⊢ (𝒫1C = (A ∪ B) → C = ∪(A ∪ B))
4		ssun1 3427	. . . . . 6 ⊢ A ⊆ (A ∪ B)
5		sseq2 3294	. . . . . 6 ⊢ (𝒫1C = (A ∪ B) → (A ⊆ 𝒫1C ↔ A ⊆ (A ∪ B)))
6	4, 5	mpbiri 224	. . . . 5 ⊢ (𝒫1C = (A ∪ B) → A ⊆ 𝒫1C)
7		pw1ss1c 4159	. . . . 5 ⊢ 𝒫1C ⊆ 1c
8	6, 7	syl6ss 3285	. . . 4 ⊢ (𝒫1C = (A ∪ B) → A ⊆ 1c)
9		eqpw1uni 4331	. . . 4 ⊢ (A ⊆ 1c → A = 𝒫1∪A)
10	8, 9	syl 15	. . 3 ⊢ (𝒫1C = (A ∪ B) → A = 𝒫1∪A)
11		ssun2 3428	. . . . . 6 ⊢ B ⊆ (A ∪ B)
12		sseq2 3294	. . . . . 6 ⊢ (𝒫1C = (A ∪ B) → (B ⊆ 𝒫1C ↔ B ⊆ (A ∪ B)))
13	11, 12	mpbiri 224	. . . . 5 ⊢ (𝒫1C = (A ∪ B) → B ⊆ 𝒫1C)
14	13, 7	syl6ss 3285	. . . 4 ⊢ (𝒫1C = (A ∪ B) → B ⊆ 1c)
15		eqpw1uni 4331	. . . 4 ⊢ (B ⊆ 1c → B = 𝒫1∪B)
16	14, 15	syl 15	. . 3 ⊢ (𝒫1C = (A ∪ B) → B = 𝒫1∪B)
17		pw1equn.1	. . . . 5 ⊢ A ∈ V
18	17	uniex 4318	. . . 4 ⊢ ∪A ∈ V
19		pw1equn.2	. . . . 5 ⊢ B ∈ V
20	19	uniex 4318	. . . 4 ⊢ ∪B ∈ V
21		uneq12 3414	. . . . . . 7 ⊢ ((x = ∪A ∧ y = ∪B) → (x ∪ y) = (∪A ∪ ∪B))
22		uniun 3911	. . . . . . 7 ⊢ ∪(A ∪ B) = (∪A ∪ ∪B)
23	21, 22	syl6eqr 2403	. . . . . 6 ⊢ ((x = ∪A ∧ y = ∪B) → (x ∪ y) = ∪(A ∪ B))
24	23	eqeq2d 2364	. . . . 5 ⊢ ((x = ∪A ∧ y = ∪B) → (C = (x ∪ y) ↔ C = ∪(A ∪ B)))
25		pw1eq 4144	. . . . . . 7 ⊢ (x = ∪A → 𝒫1x = 𝒫1∪A)
26	25	eqeq2d 2364	. . . . . 6 ⊢ (x = ∪A → (A = 𝒫1x ↔ A = 𝒫1∪A))
27	26	adantr 451	. . . . 5 ⊢ ((x = ∪A ∧ y = ∪B) → (A = 𝒫1x ↔ A = 𝒫1∪A))
28		pw1eq 4144	. . . . . . 7 ⊢ (y = ∪B → 𝒫1y = 𝒫1∪B)
29	28	eqeq2d 2364	. . . . . 6 ⊢ (y = ∪B → (B = 𝒫1y ↔ B = 𝒫1∪B))
30	29	adantl 452	. . . . 5 ⊢ ((x = ∪A ∧ y = ∪B) → (B = 𝒫1y ↔ B = 𝒫1∪B))
31	24, 27, 30	3anbi123d 1252	. . . 4 ⊢ ((x = ∪A ∧ y = ∪B) → ((C = (x ∪ y) ∧ A = 𝒫1x ∧ B = 𝒫1y) ↔ (C = ∪(A ∪ B) ∧ A = 𝒫1∪A ∧ B = 𝒫1∪B)))
32	18, 20, 31	spc2ev 2948	. . 3 ⊢ ((C = ∪(A ∪ B) ∧ A = 𝒫1∪A ∧ B = 𝒫1∪B) → ∃x∃y(C = (x ∪ y) ∧ A = 𝒫1x ∧ B = 𝒫1y))
33	3, 10, 16, 32	syl3anc 1182	. 2 ⊢ (𝒫1C = (A ∪ B) → ∃x∃y(C = (x ∪ y) ∧ A = 𝒫1x ∧ B = 𝒫1y))
34		pw1un 4164	. . . 4 ⊢ 𝒫1(x ∪ y) = (𝒫1x ∪ 𝒫1y)
35		pw1eq 4144	. . . . . 6 ⊢ (C = (x ∪ y) → 𝒫1C = 𝒫1(x ∪ y))
36		uneq12 3414	. . . . . 6 ⊢ ((A = 𝒫1x ∧ B = 𝒫1y) → (A ∪ B) = (𝒫1x ∪ 𝒫1y))
37	35, 36	eqeqan12d 2368	. . . . 5 ⊢ ((C = (x ∪ y) ∧ (A = 𝒫1x ∧ B = 𝒫1y)) → (𝒫1C = (A ∪ B) ↔ 𝒫1(x ∪ y) = (𝒫1x ∪ 𝒫1y)))
38	37	3impb 1147	. . . 4 ⊢ ((C = (x ∪ y) ∧ A = 𝒫1x ∧ B = 𝒫1y) → (𝒫1C = (A ∪ B) ↔ 𝒫1(x ∪ y) = (𝒫1x ∪ 𝒫1y)))
39	34, 38	mpbiri 224	. . 3 ⊢ ((C = (x ∪ y) ∧ A = 𝒫1x ∧ B = 𝒫1y) → 𝒫1C = (A ∪ B))
40	39	exlimivv 1635	. 2 ⊢ (∃x∃y(C = (x ∪ y) ∧ A = 𝒫1x ∧ B = 𝒫1y) → 𝒫1C = (A ∪ B))
41	33, 40	impbii 180	1 ⊢ (𝒫1C = (A ∪ B) ↔ ∃x∃y(C = (x ∪ y) ∧ A = 𝒫1x ∧ B = 𝒫1y))

This proof depends on syntax axioms:   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ∪ cun 3208   ⊆ wss 3258  ∪cuni 3892  1_cc1c 4135  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-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-typlower 4087  ax-sn 4088

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  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-ral 2620  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-pr 3743  df-uni 3893  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-imak 4190  df-p6 4192  df-sik 4193  df-ssetk 4194

This theorem is used by:  taddc  6230  letc  6232