Theorem enpw1lem1 6062

Description: Lemma for enpw1 6063. Set up stratification for the reverse direction. (Contributed by SF, 26-Feb-2015.)

Assertion

Ref	Expression
enpw1lem1	⊢ {⟨x, y⟩ ∣ {x}g{y}} ∈ V

Distinct variable group:   x,g,y

Proof of Theorem enpw1lem1

Dummy variables a b p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2863	. . . . . 6 ⊢ x ∈ V
2		vex 2863	. . . . . 6 ⊢ y ∈ V
3	1, 2	opex 4589	. . . . 5 ⊢ ⟨x, y⟩ ∈ V
4	3	eluni1 4174	. . . 4 ⊢ (⟨x, y⟩ ∈ ⋃1((( SI ◡1st ∘ 1st ) ∩ ( SI ◡2nd ∘ 2nd )) “ g) ↔ {⟨x, y⟩} ∈ ((( SI ◡1st ∘ 1st ) ∩ ( SI ◡2nd ∘ 2nd )) “ g))
5		elima 4755	. . . . 5 ⊢ ({⟨x, y⟩} ∈ ((( SI ◡1st ∘ 1st ) ∩ ( SI ◡2nd ∘ 2nd )) “ g) ↔ ∃p ∈ g p(( SI ◡1st ∘ 1st ) ∩ ( SI ◡2nd ∘ 2nd )){⟨x, y⟩})
6		brin 4694	. . . . . . 7 ⊢ (p(( SI ◡1st ∘ 1st ) ∩ ( SI ◡2nd ∘ 2nd )){⟨x, y⟩} ↔ (p( SI ◡1st ∘ 1st ){⟨x, y⟩} ∧ p( SI ◡2nd ∘ 2nd ){⟨x, y⟩}))
7		brco 4884	. . . . . . . . 9 ⊢ (p( SI ◡1st ∘ 1st ){⟨x, y⟩} ↔ ∃a(p1st a ∧ a SI ◡1st {⟨x, y⟩}))
8		ancom 437	. . . . . . . . . . 11 ⊢ ((p1st a ∧ a SI ◡1st {⟨x, y⟩}) ↔ (a SI ◡1st {⟨x, y⟩} ∧ p1st a))
9	3	brsnsi2 5777	. . . . . . . . . . . . 13 ⊢ (a SI ◡1st {⟨x, y⟩} ↔ ∃b(a = {b} ∧ b◡1st ⟨x, y⟩))
10		ancom 437	. . . . . . . . . . . . . . 15 ⊢ ((a = {b} ∧ b◡1st ⟨x, y⟩) ↔ (b◡1st ⟨x, y⟩ ∧ a = {b}))
11		brcnv 4893	. . . . . . . . . . . . . . . . 17 ⊢ (b◡1st ⟨x, y⟩ ↔ ⟨x, y⟩1st b)
12	1, 2	opbr1st 5502	. . . . . . . . . . . . . . . . 17 ⊢ (⟨x, y⟩1st b ↔ x = b)
13		equcom 1680	. . . . . . . . . . . . . . . . 17 ⊢ (x = b ↔ b = x)
14	11, 12, 13	3bitri 262	. . . . . . . . . . . . . . . 16 ⊢ (b◡1st ⟨x, y⟩ ↔ b = x)
15	14	anbi1i 676	. . . . . . . . . . . . . . 15 ⊢ ((b◡1st ⟨x, y⟩ ∧ a = {b}) ↔ (b = x ∧ a = {b}))
16	10, 15	bitri 240	. . . . . . . . . . . . . 14 ⊢ ((a = {b} ∧ b◡1st ⟨x, y⟩) ↔ (b = x ∧ a = {b}))
17	16	exbii 1582	. . . . . . . . . . . . 13 ⊢ (∃b(a = {b} ∧ b◡1st ⟨x, y⟩) ↔ ∃b(b = x ∧ a = {b}))
18		sneq 3745	. . . . . . . . . . . . . . 15 ⊢ (b = x → {b} = {x})
19	18	eqeq2d 2364	. . . . . . . . . . . . . 14 ⊢ (b = x → (a = {b} ↔ a = {x}))
20	1, 19	ceqsexv 2895	. . . . . . . . . . . . 13 ⊢ (∃b(b = x ∧ a = {b}) ↔ a = {x})
21	9, 17, 20	3bitri 262	. . . . . . . . . . . 12 ⊢ (a SI ◡1st {⟨x, y⟩} ↔ a = {x})
22	21	anbi1i 676	. . . . . . . . . . 11 ⊢ ((a SI ◡1st {⟨x, y⟩} ∧ p1st a) ↔ (a = {x} ∧ p1st a))
23	8, 22	bitri 240	. . . . . . . . . 10 ⊢ ((p1st a ∧ a SI ◡1st {⟨x, y⟩}) ↔ (a = {x} ∧ p1st a))
24	23	exbii 1582	. . . . . . . . 9 ⊢ (∃a(p1st a ∧ a SI ◡1st {⟨x, y⟩}) ↔ ∃a(a = {x} ∧ p1st a))
25		snex 4112	. . . . . . . . . 10 ⊢ {x} ∈ V
26		breq2 4644	. . . . . . . . . 10 ⊢ (a = {x} → (p1st a ↔ p1st {x}))
27	25, 26	ceqsexv 2895	. . . . . . . . 9 ⊢ (∃a(a = {x} ∧ p1st a) ↔ p1st {x})
28	7, 24, 27	3bitri 262	. . . . . . . 8 ⊢ (p( SI ◡1st ∘ 1st ){⟨x, y⟩} ↔ p1st {x})
29		brco 4884	. . . . . . . . 9 ⊢ (p( SI ◡2nd ∘ 2nd ){⟨x, y⟩} ↔ ∃a(p2nd a ∧ a SI ◡2nd {⟨x, y⟩}))
30	3	brsnsi2 5777	. . . . . . . . . . . . 13 ⊢ (a SI ◡2nd {⟨x, y⟩} ↔ ∃b(a = {b} ∧ b◡2nd ⟨x, y⟩))
31		brcnv 4893	. . . . . . . . . . . . . . . . 17 ⊢ (b◡2nd ⟨x, y⟩ ↔ ⟨x, y⟩2nd b)
32	1, 2	opbr2nd 5503	. . . . . . . . . . . . . . . . 17 ⊢ (⟨x, y⟩2nd b ↔ y = b)
33		equcom 1680	. . . . . . . . . . . . . . . . 17 ⊢ (y = b ↔ b = y)
34	31, 32, 33	3bitri 262	. . . . . . . . . . . . . . . 16 ⊢ (b◡2nd ⟨x, y⟩ ↔ b = y)
35	34	anbi2i 675	. . . . . . . . . . . . . . 15 ⊢ ((a = {b} ∧ b◡2nd ⟨x, y⟩) ↔ (a = {b} ∧ b = y))
36		ancom 437	. . . . . . . . . . . . . . 15 ⊢ ((a = {b} ∧ b = y) ↔ (b = y ∧ a = {b}))
37	35, 36	bitri 240	. . . . . . . . . . . . . 14 ⊢ ((a = {b} ∧ b◡2nd ⟨x, y⟩) ↔ (b = y ∧ a = {b}))
38	37	exbii 1582	. . . . . . . . . . . . 13 ⊢ (∃b(a = {b} ∧ b◡2nd ⟨x, y⟩) ↔ ∃b(b = y ∧ a = {b}))
39		sneq 3745	. . . . . . . . . . . . . . 15 ⊢ (b = y → {b} = {y})
40	39	eqeq2d 2364	. . . . . . . . . . . . . 14 ⊢ (b = y → (a = {b} ↔ a = {y}))
41	2, 40	ceqsexv 2895	. . . . . . . . . . . . 13 ⊢ (∃b(b = y ∧ a = {b}) ↔ a = {y})
42	30, 38, 41	3bitri 262	. . . . . . . . . . . 12 ⊢ (a SI ◡2nd {⟨x, y⟩} ↔ a = {y})
43	42	anbi2i 675	. . . . . . . . . . 11 ⊢ ((p2nd a ∧ a SI ◡2nd {⟨x, y⟩}) ↔ (p2nd a ∧ a = {y}))
44		ancom 437	. . . . . . . . . . 11 ⊢ ((p2nd a ∧ a = {y}) ↔ (a = {y} ∧ p2nd a))
45	43, 44	bitri 240	. . . . . . . . . 10 ⊢ ((p2nd a ∧ a SI ◡2nd {⟨x, y⟩}) ↔ (a = {y} ∧ p2nd a))
46	45	exbii 1582	. . . . . . . . 9 ⊢ (∃a(p2nd a ∧ a SI ◡2nd {⟨x, y⟩}) ↔ ∃a(a = {y} ∧ p2nd a))
47		snex 4112	. . . . . . . . . 10 ⊢ {y} ∈ V
48		breq2 4644	. . . . . . . . . 10 ⊢ (a = {y} → (p2nd a ↔ p2nd {y}))
49	47, 48	ceqsexv 2895	. . . . . . . . 9 ⊢ (∃a(a = {y} ∧ p2nd a) ↔ p2nd {y})
50	29, 46, 49	3bitri 262	. . . . . . . 8 ⊢ (p( SI ◡2nd ∘ 2nd ){⟨x, y⟩} ↔ p2nd {y})
51	28, 50	anbi12i 678	. . . . . . 7 ⊢ ((p( SI ◡1st ∘ 1st ){⟨x, y⟩} ∧ p( SI ◡2nd ∘ 2nd ){⟨x, y⟩}) ↔ (p1st {x} ∧ p2nd {y}))
52	25, 47	op1st2nd 5791	. . . . . . 7 ⊢ ((p1st {x} ∧ p2nd {y}) ↔ p = ⟨{x}, {y}⟩)
53	6, 51, 52	3bitri 262	. . . . . 6 ⊢ (p(( SI ◡1st ∘ 1st ) ∩ ( SI ◡2nd ∘ 2nd )){⟨x, y⟩} ↔ p = ⟨{x}, {y}⟩)
54	53	rexbii 2640	. . . . 5 ⊢ (∃p ∈ g p(( SI ◡1st ∘ 1st ) ∩ ( SI ◡2nd ∘ 2nd )){⟨x, y⟩} ↔ ∃p ∈ g p = ⟨{x}, {y}⟩)
55	5, 54	bitri 240	. . . 4 ⊢ ({⟨x, y⟩} ∈ ((( SI ◡1st ∘ 1st ) ∩ ( SI ◡2nd ∘ 2nd )) “ g) ↔ ∃p ∈ g p = ⟨{x}, {y}⟩)
56		df-br 4641	. . . . 5 ⊢ ({x}g{y} ↔ ⟨{x}, {y}⟩ ∈ g)
57		risset 2662	. . . . 5 ⊢ (⟨{x}, {y}⟩ ∈ g ↔ ∃p ∈ g p = ⟨{x}, {y}⟩)
58	56, 57	bitr2i 241	. . . 4 ⊢ (∃p ∈ g p = ⟨{x}, {y}⟩ ↔ {x}g{y})
59	4, 55, 58	3bitri 262	. . 3 ⊢ (⟨x, y⟩ ∈ ⋃1((( SI ◡1st ∘ 1st ) ∩ ( SI ◡2nd ∘ 2nd )) “ g) ↔ {x}g{y})
60	59	opabbi2i 4867	. 2 ⊢ ⋃1((( SI ◡1st ∘ 1st ) ∩ ( SI ◡2nd ∘ 2nd )) “ g) = {⟨x, y⟩ ∣ {x}g{y}}
61		1stex 4740	. . . . . . . 8 ⊢ 1st ∈ V
62	61	cnvex 5103	. . . . . . 7 ⊢ ◡1st ∈ V
63	62	siex 4754	. . . . . 6 ⊢ SI ◡1st ∈ V
64	63, 61	coex 4751	. . . . 5 ⊢ ( SI ◡1st ∘ 1st ) ∈ V
65		2ndex 5113	. . . . . . . 8 ⊢ 2nd ∈ V
66	65	cnvex 5103	. . . . . . 7 ⊢ ◡2nd ∈ V
67	66	siex 4754	. . . . . 6 ⊢ SI ◡2nd ∈ V
68	67, 65	coex 4751	. . . . 5 ⊢ ( SI ◡2nd ∘ 2nd ) ∈ V
69	64, 68	inex 4106	. . . 4 ⊢ (( SI ◡1st ∘ 1st ) ∩ ( SI ◡2nd ∘ 2nd )) ∈ V
70		vex 2863	. . . 4 ⊢ g ∈ V
71	69, 70	imaex 4748	. . 3 ⊢ ((( SI ◡1st ∘ 1st ) ∩ ( SI ◡2nd ∘ 2nd )) “ g) ∈ V
72	71	uni1ex 4294	. 2 ⊢ ⋃1((( SI ◡1st ∘ 1st ) ∩ ( SI ◡2nd ∘ 2nd )) “ g) ∈ V
73	60, 72	eqeltrri 2424	1 ⊢ {⟨x, y⟩ ∣ {x}g{y}} ∈ V

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  Vcvv 2860   ∩ cin 3209  {csn 3738  ⋃₁cuni1 4134  ⟨cop 4562  {copab 4623   class class class wbr 4640  1^st c1st 4718   SI csi 4721   ∘ ccom 4722   “ cima 4723  ^◡ccnv 4772  2^nd c2nd 4784

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-13 1712  ax-14 1714  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-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-co 4727  df-ima 4728  df-si 4729  df-cnv 4786  df-2nd 4798

This theorem is referenced by:  enpw1  6063