Theorem pw1equn 4331

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	⊢ (℘1C = (A ∪ B) ↔ ∃x∃y(C = (x ∪ y) ∧ A = ℘1x ∧ B = ℘1y))

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

Proof of Theorem pw1equn

Step	Hyp	Ref	Expression
1		unipw1 4325	. . . 4 ⊢ ∪℘1C = C
2		unieq 3900	. . . 4 ⊢ (℘1C = (A ∪ B) → ∪℘1C = ∪(A ∪ B))
3	1, 2	syl5eqr 2399	. . 3 ⊢ (℘1C = (A ∪ B) → C = ∪(A ∪ B))
4		ssun1 3426	. . . . . 6 ⊢ A ⊆ (A ∪ B)
5		sseq2 3293	. . . . . 6 ⊢ (℘1C = (A ∪ B) → (A ⊆ ℘1C ↔ A ⊆ (A ∪ B)))
6	4, 5	mpbiri 224	. . . . 5 ⊢ (℘1C = (A ∪ B) → A ⊆ ℘1C)
7		pw1ss1c 4158	. . . . 5 ⊢ ℘1C ⊆ 1c
8	6, 7	syl6ss 3284	. . . 4 ⊢ (℘1C = (A ∪ B) → A ⊆ 1c)
9		eqpw1uni 4330	. . . 4 ⊢ (A ⊆ 1c → A = ℘1∪A)
10	8, 9	syl 15	. . 3 ⊢ (℘1C = (A ∪ B) → A = ℘1∪A)
11		ssun2 3427	. . . . . 6 ⊢ B ⊆ (A ∪ B)
12		sseq2 3293	. . . . . 6 ⊢ (℘1C = (A ∪ B) → (B ⊆ ℘1C ↔ B ⊆ (A ∪ B)))
13	11, 12	mpbiri 224	. . . . 5 ⊢ (℘1C = (A ∪ B) → B ⊆ ℘1C)
14	13, 7	syl6ss 3284	. . . 4 ⊢ (℘1C = (A ∪ B) → B ⊆ 1c)
15		eqpw1uni 4330	. . . 4 ⊢ (B ⊆ 1c → B = ℘1∪B)
16	14, 15	syl 15	. . 3 ⊢ (℘1C = (A ∪ B) → B = ℘1∪B)
17		pw1equn.1	. . . . 5 ⊢ A ∈ V
18	17	uniex 4317	. . . 4 ⊢ ∪A ∈ V
19		pw1equn.2	. . . . 5 ⊢ B ∈ V
20	19	uniex 4317	. . . 4 ⊢ ∪B ∈ V
21		uneq12 3413	. . . . . . 7 ⊢ ((x = ∪A ∧ y = ∪B) → (x ∪ y) = (∪A ∪ ∪B))
22		uniun 3910	. . . . . . 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 4143	. . . . . . 7 ⊢ (x = ∪A → ℘1x = ℘1∪A)
26	25	eqeq2d 2364	. . . . . 6 ⊢ (x = ∪A → (A = ℘1x ↔ A = ℘1∪A))
27	26	adantr 451	. . . . 5 ⊢ ((x = ∪A ∧ y = ∪B) → (A = ℘1x ↔ A = ℘1∪A))
28		pw1eq 4143	. . . . . . 7 ⊢ (y = ∪B → ℘1y = ℘1∪B)
29	28	eqeq2d 2364	. . . . . 6 ⊢ (y = ∪B → (B = ℘1y ↔ B = ℘1∪B))
30	29	adantl 452	. . . . 5 ⊢ ((x = ∪A ∧ y = ∪B) → (B = ℘1y ↔ B = ℘1∪B))
31	24, 27, 30	3anbi123d 1252	. . . 4 ⊢ ((x = ∪A ∧ y = ∪B) → ((C = (x ∪ y) ∧ A = ℘1x ∧ B = ℘1y) ↔ (C = ∪(A ∪ B) ∧ A = ℘1∪A ∧ B = ℘1∪B)))
32	18, 20, 31	spc2ev 2947	. . 3 ⊢ ((C = ∪(A ∪ B) ∧ A = ℘1∪A ∧ B = ℘1∪B) → ∃x∃y(C = (x ∪ y) ∧ A = ℘1x ∧ B = ℘1y))
33	3, 10, 16, 32	syl3anc 1182	. 2 ⊢ (℘1C = (A ∪ B) → ∃x∃y(C = (x ∪ y) ∧ A = ℘1x ∧ B = ℘1y))
34		pw1un 4163	. . . 4 ⊢ ℘1(x ∪ y) = (℘1x ∪ ℘1y)
35		pw1eq 4143	. . . . . 6 ⊢ (C = (x ∪ y) → ℘1C = ℘1(x ∪ y))
36		uneq12 3413	. . . . . 6 ⊢ ((A = ℘1x ∧ B = ℘1y) → (A ∪ B) = (℘1x ∪ ℘1y))
37	35, 36	eqeqan12d 2368	. . . . 5 ⊢ ((C = (x ∪ y) ∧ (A = ℘1x ∧ B = ℘1y)) → (℘1C = (A ∪ B) ↔ ℘1(x ∪ y) = (℘1x ∪ ℘1y)))
38	37	3impb 1147	. . . 4 ⊢ ((C = (x ∪ y) ∧ A = ℘1x ∧ B = ℘1y) → (℘1C = (A ∪ B) ↔ ℘1(x ∪ y) = (℘1x ∪ ℘1y)))
39	34, 38	mpbiri 224	. . 3 ⊢ ((C = (x ∪ y) ∧ A = ℘1x ∧ B = ℘1y) → ℘1C = (A ∪ B))
40	39	exlimivv 1635	. 2 ⊢ (∃x∃y(C = (x ∪ y) ∧ A = ℘1x ∧ B = ℘1y) → ℘1C = (A ∪ B))
41	33, 40	impbii 180	1 ⊢ (℘1C = (A ∪ B) ↔ ∃x∃y(C = (x ∪ y) ∧ A = ℘1x ∧ B = ℘1y))

Syntax hints:   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ∪ cun 3207   ⊆ wss 3257  ∪cuni 3891  1_cc1c 4134  ℘₁cpw1 4135

This theorem was proved from 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-typlower 4086  ax-sn 4087

This theorem 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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-imak 4189  df-p6 4191  df-sik 4192  df-ssetk 4193

This theorem is referenced by:  taddc  6229  letc  6231