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}} e. _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 e. _V
2		vex 2863	. . . . . 6 |- y e. _V
3	1, 2	opex 4589	. . . . 5 |- <.x, y>. e. _V
4	3	eluni1 4174	. . . 4 |- (<.x, y>. e. ⋃1((( SI `'1st o. 1st) i^i ( SI `'2nd o. 2nd))"g) <-> {<.x, y>.} e. ((( SI `'1st o. 1st) i^i ( SI `'2nd o. 2nd))"g))
5		elima 4755	. . . . 5 |- ({<.x, y>.} e. ((( SI `'1st o. 1st) i^i ( SI `'2nd o. 2nd))"g) <-> E.p e. g p(( SI `'1st o. 1st) i^i ( SI `'2nd o. 2nd)){<.x, y>.})
6		brin 4694	. . . . . . 7 |- (p(( SI `'1st o. 1st) i^i ( SI `'2nd o. 2nd)){<.x, y>.} <-> (p( SI `'1st o. 1st){<.x, y>.} /\ p( SI `'2nd o. 2nd){<.x, y>.}))
7		brco 4884	. . . . . . . . 9 |- (p( SI `'1st o. 1st){<.x, y>.} <-> E.a(p1sta /\ a SI `'1st{<.x, y>.}))
8		ancom 437	. . . . . . . . . . 11 |- ((p1sta /\ a SI `'1st{<.x, y>.}) <-> (a SI `'1st{<.x, y>.} /\ p1sta))
9	3	brsnsi2 5777	. . . . . . . . . . . . 13 |- (a SI `'1st{<.x, y>.} <-> E.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>.1stb)
12	1, 2	opbr1st 5502	. . . . . . . . . . . . . . . . 17 |- (<.x, y>.1stb <-> 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 |- (E.b(a = {b} /\ b`'1st<.x, y>.) <-> E.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 |- (E.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>.} /\ p1sta) <-> (a = {x} /\ p1sta))
23	8, 22	bitri 240	. . . . . . . . . 10 |- ((p1sta /\ a SI `'1st{<.x, y>.}) <-> (a = {x} /\ p1sta))
24	23	exbii 1582	. . . . . . . . 9 |- (E.a(p1sta /\ a SI `'1st{<.x, y>.}) <-> E.a(a = {x} /\ p1sta))
25		snex 4112	. . . . . . . . . 10 |- {x} e. _V
26		breq2 4644	. . . . . . . . . 10 |- (a = {x} -> (p1sta <-> p1st{x}))
27	25, 26	ceqsexv 2895	. . . . . . . . 9 |- (E.a(a = {x} /\ p1sta) <-> p1st{x})
28	7, 24, 27	3bitri 262	. . . . . . . 8 |- (p( SI `'1st o. 1st){<.x, y>.} <-> p1st{x})
29		brco 4884	. . . . . . . . 9 |- (p( SI `'2nd o. 2nd){<.x, y>.} <-> E.a(p2nda /\ a SI `'2nd{<.x, y>.}))
30	3	brsnsi2 5777	. . . . . . . . . . . . 13 |- (a SI `'2nd{<.x, y>.} <-> E.b(a = {b} /\ b`'2nd<.x, y>.))
31		brcnv 4893	. . . . . . . . . . . . . . . . 17 |- (b`'2nd<.x, y>. <-> <.x, y>.2ndb)
32	1, 2	opbr2nd 5503	. . . . . . . . . . . . . . . . 17 |- (<.x, y>.2ndb <-> 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 |- (E.b(a = {b} /\ b`'2nd<.x, y>.) <-> E.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 |- (E.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 |- ((p2nda /\ a SI `'2nd{<.x, y>.}) <-> (p2nda /\ a = {y}))
44		ancom 437	. . . . . . . . . . 11 |- ((p2nda /\ a = {y}) <-> (a = {y} /\ p2nda))
45	43, 44	bitri 240	. . . . . . . . . 10 |- ((p2nda /\ a SI `'2nd{<.x, y>.}) <-> (a = {y} /\ p2nda))
46	45	exbii 1582	. . . . . . . . 9 |- (E.a(p2nda /\ a SI `'2nd{<.x, y>.}) <-> E.a(a = {y} /\ p2nda))
47		snex 4112	. . . . . . . . . 10 |- {y} e. _V
48		breq2 4644	. . . . . . . . . 10 |- (a = {y} -> (p2nda <-> p2nd{y}))
49	47, 48	ceqsexv 2895	. . . . . . . . 9 |- (E.a(a = {y} /\ p2nda) <-> p2nd{y})
50	29, 46, 49	3bitri 262	. . . . . . . 8 |- (p( SI `'2nd o. 2nd){<.x, y>.} <-> p2nd{y})
51	28, 50	anbi12i 678	. . . . . . 7 |- ((p( SI `'1st o. 1st){<.x, y>.} /\ p( SI `'2nd o. 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 o. 1st) i^i ( SI `'2nd o. 2nd)){<.x, y>.} <-> p = <.{x}, {y}>.)
54	53	rexbii 2640	. . . . 5 |- (E.p e. g p(( SI `'1st o. 1st) i^i ( SI `'2nd o. 2nd)){<.x, y>.} <-> E.p e. g p = <.{x}, {y}>.)
55	5, 54	bitri 240	. . . 4 |- ({<.x, y>.} e. ((( SI `'1st o. 1st) i^i ( SI `'2nd o. 2nd))"g) <-> E.p e. g p = <.{x}, {y}>.)
56		df-br 4641	. . . . 5 |- ({x}g{y} <-> <.{x}, {y}>. e. g)
57		risset 2662	. . . . 5 |- (<.{x}, {y}>. e. g <-> E.p e. g p = <.{x}, {y}>.)
58	56, 57	bitr2i 241	. . . 4 |- (E.p e. g p = <.{x}, {y}>. <-> {x}g{y})
59	4, 55, 58	3bitri 262	. . 3 |- (<.x, y>. e. ⋃1((( SI `'1st o. 1st) i^i ( SI `'2nd o. 2nd))"g) <-> {x}g{y})
60	59	opabbi2i 4867	. 2 |- ⋃1((( SI `'1st o. 1st) i^i ( SI `'2nd o. 2nd))"g) = {<.x, y>. | {x}g{y}}
61		1stex 4740	. . . . . . . 8 |- 1st e. _V
62	61	cnvex 5103	. . . . . . 7 |- `'1st e. _V
63	62	siex 4754	. . . . . 6 |- SI `'1st e. _V
64	63, 61	coex 4751	. . . . 5 |- ( SI `'1st o. 1st) e. _V
65		2ndex 5113	. . . . . . . 8 |- 2nd e. _V
66	65	cnvex 5103	. . . . . . 7 |- `'2nd e. _V
67	66	siex 4754	. . . . . 6 |- SI `'2nd e. _V
68	67, 65	coex 4751	. . . . 5 |- ( SI `'2nd o. 2nd) e. _V
69	64, 68	inex 4106	. . . 4 |- (( SI `'1st o. 1st) i^i ( SI `'2nd o. 2nd)) e. _V
70		vex 2863	. . . 4 |- g e. _V
71	69, 70	imaex 4748	. . 3 |- ((( SI `'1st o. 1st) i^i ( SI `'2nd o. 2nd))"g) e. _V
72	71	uni1ex 4294	. 2 |- ⋃1((( SI `'1st o. 1st) i^i ( SI `'2nd o. 2nd))"g) e. _V
73	60, 72	eqeltrri 2424	1 |- {<.x, y>. | {x}g{y}} e. _V

Syntax hints:   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  E.wrex 2616  _Vcvv 2860   i^i cin 3209  {csn 3738  ⋃₁cuni1 4134  <.cop 4562  {copab 4623   class class class wbr 4640  1stc1st 4718   SI csi 4721   o. ccom 4722  "cima 4723  `'ccnv 4772  2ndc2nd 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