Theorem pw1eqadj 4332

Description: A condition for a unit power class to work out to an adjunction. (Contributed by SF, 26-Jan-2015.)

Hypotheses

Ref	Expression
pw1eqadj.1	⊢ A ∈ V
pw1eqadj.2	⊢ B ∈ V

Assertion

Ref	Expression
pw1eqadj	⊢ (℘1C = (A ∪ {B}) ↔ ∃x∃y(C = (x ∪ {y}) ∧ A = ℘1x ∧ B = {y}))

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

Proof of Theorem pw1eqadj

Step	Hyp	Ref	Expression
1		unieq 3900	. . . . 5 ⊢ (℘1C = (A ∪ {B}) → ∪℘1C = ∪(A ∪ {B}))
2		unipw1 4325	. . . . 5 ⊢ ∪℘1C = C
3		uniun 3910	. . . . 5 ⊢ ∪(A ∪ {B}) = (∪A ∪ ∪{B})
4	1, 2, 3	3eqtr3g 2408	. . . 4 ⊢ (℘1C = (A ∪ {B}) → C = (∪A ∪ ∪{B}))
5		pw1eqadj.2	. . . . . . 7 ⊢ B ∈ V
6	5	unisn 3907	. . . . . 6 ⊢ ∪{B} = B
7		pw1ss1c 4158	. . . . . . . 8 ⊢ ℘1C ⊆ 1c
8		ssun2 3427	. . . . . . . . . 10 ⊢ {B} ⊆ (A ∪ {B})
9	5	snid 3760	. . . . . . . . . 10 ⊢ B ∈ {B}
10	8, 9	sselii 3270	. . . . . . . . 9 ⊢ B ∈ (A ∪ {B})
11		eleq2 2414	. . . . . . . . 9 ⊢ (℘1C = (A ∪ {B}) → (B ∈ ℘1C ↔ B ∈ (A ∪ {B})))
12	10, 11	mpbiri 224	. . . . . . . 8 ⊢ (℘1C = (A ∪ {B}) → B ∈ ℘1C)
13	7, 12	sseldi 3271	. . . . . . 7 ⊢ (℘1C = (A ∪ {B}) → B ∈ 1c)
14		el1c 4139	. . . . . . . 8 ⊢ (B ∈ 1c ↔ ∃x B = {x})
15		vex 2862	. . . . . . . . . . . . 13 ⊢ x ∈ V
16	15	unisn 3907	. . . . . . . . . . . 12 ⊢ ∪{x} = x
17	16	sneqi 3745	. . . . . . . . . . 11 ⊢ {∪{x}} = {x}
18	17	eqcomi 2357	. . . . . . . . . 10 ⊢ {x} = {∪{x}}
19		id 19	. . . . . . . . . 10 ⊢ (B = {x} → B = {x})
20		unieq 3900	. . . . . . . . . . 11 ⊢ (B = {x} → ∪B = ∪{x})
21	20	sneqd 3746	. . . . . . . . . 10 ⊢ (B = {x} → {∪B} = {∪{x}})
22	18, 19, 21	3eqtr4a 2411	. . . . . . . . 9 ⊢ (B = {x} → B = {∪B})
23	22	exlimiv 1634	. . . . . . . 8 ⊢ (∃x B = {x} → B = {∪B})
24	14, 23	sylbi 187	. . . . . . 7 ⊢ (B ∈ 1c → B = {∪B})
25	13, 24	syl 15	. . . . . 6 ⊢ (℘1C = (A ∪ {B}) → B = {∪B})
26	6, 25	syl5eq 2397	. . . . 5 ⊢ (℘1C = (A ∪ {B}) → ∪{B} = {∪B})
27	26	uneq2d 3418	. . . 4 ⊢ (℘1C = (A ∪ {B}) → (∪A ∪ ∪{B}) = (∪A ∪ {∪B}))
28	4, 27	eqtrd 2385	. . 3 ⊢ (℘1C = (A ∪ {B}) → C = (∪A ∪ {∪B}))
29		ssun1 3426	. . . . . 6 ⊢ A ⊆ (A ∪ {B})
30		sseq2 3293	. . . . . 6 ⊢ (℘1C = (A ∪ {B}) → (A ⊆ ℘1C ↔ A ⊆ (A ∪ {B})))
31	29, 30	mpbiri 224	. . . . 5 ⊢ (℘1C = (A ∪ {B}) → A ⊆ ℘1C)
32	31, 7	syl6ss 3284	. . . 4 ⊢ (℘1C = (A ∪ {B}) → A ⊆ 1c)
33		eqpw1uni 4330	. . . 4 ⊢ (A ⊆ 1c → A = ℘1∪A)
34	32, 33	syl 15	. . 3 ⊢ (℘1C = (A ∪ {B}) → A = ℘1∪A)
35		pw1eqadj.1	. . . . 5 ⊢ A ∈ V
36	35	uniex 4317	. . . 4 ⊢ ∪A ∈ V
37	5	uniex 4317	. . . 4 ⊢ ∪B ∈ V
38		sneq 3744	. . . . . . 7 ⊢ (y = ∪B → {y} = {∪B})
39		uneq12 3413	. . . . . . 7 ⊢ ((x = ∪A ∧ {y} = {∪B}) → (x ∪ {y}) = (∪A ∪ {∪B}))
40	38, 39	sylan2 460	. . . . . 6 ⊢ ((x = ∪A ∧ y = ∪B) → (x ∪ {y}) = (∪A ∪ {∪B}))
41	40	eqeq2d 2364	. . . . 5 ⊢ ((x = ∪A ∧ y = ∪B) → (C = (x ∪ {y}) ↔ C = (∪A ∪ {∪B})))
42		pw1eq 4143	. . . . . . 7 ⊢ (x = ∪A → ℘1x = ℘1∪A)
43	42	eqeq2d 2364	. . . . . 6 ⊢ (x = ∪A → (A = ℘1x ↔ A = ℘1∪A))
44	43	adantr 451	. . . . 5 ⊢ ((x = ∪A ∧ y = ∪B) → (A = ℘1x ↔ A = ℘1∪A))
45	38	eqeq2d 2364	. . . . . 6 ⊢ (y = ∪B → (B = {y} ↔ B = {∪B}))
46	45	adantl 452	. . . . 5 ⊢ ((x = ∪A ∧ y = ∪B) → (B = {y} ↔ B = {∪B}))
47	41, 44, 46	3anbi123d 1252	. . . 4 ⊢ ((x = ∪A ∧ y = ∪B) → ((C = (x ∪ {y}) ∧ A = ℘1x ∧ B = {y}) ↔ (C = (∪A ∪ {∪B}) ∧ A = ℘1∪A ∧ B = {∪B})))
48	36, 37, 47	spc2ev 2947	. . 3 ⊢ ((C = (∪A ∪ {∪B}) ∧ A = ℘1∪A ∧ B = {∪B}) → ∃x∃y(C = (x ∪ {y}) ∧ A = ℘1x ∧ B = {y}))
49	28, 34, 25, 48	syl3anc 1182	. 2 ⊢ (℘1C = (A ∪ {B}) → ∃x∃y(C = (x ∪ {y}) ∧ A = ℘1x ∧ B = {y}))
50		pw1un 4163	. . . . 5 ⊢ ℘1(x ∪ {y}) = (℘1x ∪ ℘1{y})
51		vex 2862	. . . . . . 7 ⊢ y ∈ V
52	51	pw1sn 4165	. . . . . 6 ⊢ ℘1{y} = {{y}}
53	52	uneq2i 3415	. . . . 5 ⊢ (℘1x ∪ ℘1{y}) = (℘1x ∪ {{y}})
54	50, 53	eqtri 2373	. . . 4 ⊢ ℘1(x ∪ {y}) = (℘1x ∪ {{y}})
55		pw1eq 4143	. . . . . 6 ⊢ (C = (x ∪ {y}) → ℘1C = ℘1(x ∪ {y}))
56		sneq 3744	. . . . . . 7 ⊢ (B = {y} → {B} = {{y}})
57		uneq12 3413	. . . . . . 7 ⊢ ((A = ℘1x ∧ {B} = {{y}}) → (A ∪ {B}) = (℘1x ∪ {{y}}))
58	56, 57	sylan2 460	. . . . . 6 ⊢ ((A = ℘1x ∧ B = {y}) → (A ∪ {B}) = (℘1x ∪ {{y}}))
59	55, 58	eqeqan12d 2368	. . . . 5 ⊢ ((C = (x ∪ {y}) ∧ (A = ℘1x ∧ B = {y})) → (℘1C = (A ∪ {B}) ↔ ℘1(x ∪ {y}) = (℘1x ∪ {{y}})))
60	59	3impb 1147	. . . 4 ⊢ ((C = (x ∪ {y}) ∧ A = ℘1x ∧ B = {y}) → (℘1C = (A ∪ {B}) ↔ ℘1(x ∪ {y}) = (℘1x ∪ {{y}})))
61	54, 60	mpbiri 224	. . 3 ⊢ ((C = (x ∪ {y}) ∧ A = ℘1x ∧ B = {y}) → ℘1C = (A ∪ {B}))
62	61	exlimivv 1635	. 2 ⊢ (∃x∃y(C = (x ∪ {y}) ∧ A = ℘1x ∧ B = {y}) → ℘1C = (A ∪ {B}))
63	49, 62	impbii 180	1 ⊢ (℘1C = (A ∪ {B}) ↔ ∃x∃y(C = (x ∪ {y}) ∧ A = ℘1x ∧ B = {y}))

Syntax hints:   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ∪ cun 3207   ⊆ wss 3257  {csn 3737  ∪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-sbc 3047  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:  ncfinlower  4483  sfindbl  4530