Theorem oprabid 5551

Description: The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 20-Mar-2013.)

Assertion

Ref	Expression
oprabid	|- (<.<.x, y>., z>. e. {<.<.x, y>., z>. | ph} <-> ph)

Proof of Theorem oprabid

Dummy variables a r s t w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2863	. . . 4 |- x e. _V
2		vex 2863	. . . 4 |- y e. _V
3	1, 2	opex 4589	. . 3 |- <.x, y>. e. _V
4		vex 2863	. . 3 |- z e. _V
5	3, 4	opex 4589	. 2 |- <.<.x, y>., z>. e. _V
6	3, 4	eqvinop 4607	. . . . 5 |- (w = <.<.x, y>., z>. <-> E.aE.t(w = <.a, t>. /\ <.a, t>. = <.<.x, y>., z>.))
7	6	biimpi 186	. . . 4 |- (w = <.<.x, y>., z>. -> E.aE.t(w = <.a, t>. /\ <.a, t>. = <.<.x, y>., z>.))
8		eqeq1 2359	. . . . . . . 8 |- (w = <.a, t>. -> (w = <.<.x, y>., z>. <-> <.a, t>. = <.<.x, y>., z>.))
9		opth 4603	. . . . . . . . 9 |- (<.a, t>. = <.<.x, y>., z>. <-> (a = <.x, y>. /\ t = z))
10	9	simplbi 446	. . . . . . . 8 |- (<.a, t>. = <.<.x, y>., z>. -> a = <.x, y>.)
11	8, 10	syl6bi 219	. . . . . . 7 |- (w = <.a, t>. -> (w = <.<.x, y>., z>. -> a = <.x, y>.))
12	1, 2	eqvinop 4607	. . . . . . . . 9 |- (a = <.x, y>. <-> E.rE.s(a = <.r, s>. /\ <.r, s>. = <.x, y>.))
13		opeq1 4579	. . . . . . . . . . . . 13 |- (a = <.r, s>. -> <.a, t>. = <.<.r, s>., t>.)
14	13	eqeq2d 2364	. . . . . . . . . . . 12 |- (a = <.r, s>. -> (w = <.a, t>. <-> w = <.<.r, s>., t>.))
15		opth 4603	. . . . . . . . . . . . . . . . . . 19 |- (<.<.x, y>., z>. = <.<.r, s>., t>. <-> (<.x, y>. = <.r, s>. /\ z = t))
16		opth 4603	. . . . . . . . . . . . . . . . . . . 20 |- (<.x, y>. = <.r, s>. <-> (x = r /\ y = s))
17	16	anbi1i 676	. . . . . . . . . . . . . . . . . . 19 |- ((<.x, y>. = <.r, s>. /\ z = t) <-> ((x = r /\ y = s) /\ z = t))
18	15, 17	bitri 240	. . . . . . . . . . . . . . . . . 18 |- (<.<.x, y>., z>. = <.<.r, s>., t>. <-> ((x = r /\ y = s) /\ z = t))
19	18	anbi1i 676	. . . . . . . . . . . . . . . . 17 |- ((<.<.x, y>., z>. = <.<.r, s>., t>. /\ ph) <-> (((x = r /\ y = s) /\ z = t) /\ ph))
20		anass 630	. . . . . . . . . . . . . . . . 17 |- ((((x = r /\ y = s) /\ z = t) /\ ph) <-> ((x = r /\ y = s) /\ (z = t /\ ph)))
21		anass 630	. . . . . . . . . . . . . . . . 17 |- (((x = r /\ y = s) /\ (z = t /\ ph)) <-> (x = r /\ (y = s /\ (z = t /\ ph))))
22	19, 20, 21	3bitri 262	. . . . . . . . . . . . . . . 16 |- ((<.<.x, y>., z>. = <.<.r, s>., t>. /\ ph) <-> (x = r /\ (y = s /\ (z = t /\ ph))))
23	22	3exbii 1584	. . . . . . . . . . . . . . 15 |- (E.xE.yE.z(<.<.x, y>., z>. = <.<.r, s>., t>. /\ ph) <-> E.xE.yE.z(x = r /\ (y = s /\ (z = t /\ ph))))
24		nfcvf2 2513	. . . . . . . . . . . . . . . . . . . 20 |- (-. A.x x = z -> F/_zx)
25		nfcvd 2491	. . . . . . . . . . . . . . . . . . . 20 |- (-. A.x x = z -> F/_zr)
26	24, 25	nfeqd 2504	. . . . . . . . . . . . . . . . . . 19 |- (-. A.x x = z -> F/z x = r)
27	26	exdistrf 1971	. . . . . . . . . . . . . . . . . 18 |- (E.xE.z(x = r /\ (y = s /\ (z = t /\ ph))) -> E.x(x = r /\ E.z(y = s /\ (z = t /\ ph))))
28	27	eximi 1576	. . . . . . . . . . . . . . . . 17 |- (E.yE.xE.z(x = r /\ (y = s /\ (z = t /\ ph))) -> E.yE.x(x = r /\ E.z(y = s /\ (z = t /\ ph))))
29		excom 1741	. . . . . . . . . . . . . . . . 17 |- (E.xE.yE.z(x = r /\ (y = s /\ (z = t /\ ph))) <-> E.yE.xE.z(x = r /\ (y = s /\ (z = t /\ ph))))
30		excom 1741	. . . . . . . . . . . . . . . . 17 |- (E.xE.y(x = r /\ E.z(y = s /\ (z = t /\ ph))) <-> E.yE.x(x = r /\ E.z(y = s /\ (z = t /\ ph))))
31	28, 29, 30	3imtr4i 257	. . . . . . . . . . . . . . . 16 |- (E.xE.yE.z(x = r /\ (y = s /\ (z = t /\ ph))) -> E.xE.y(x = r /\ E.z(y = s /\ (z = t /\ ph))))
32		nfcvf2 2513	. . . . . . . . . . . . . . . . . 18 |- (-. A.x x = y -> F/_yx)
33		nfcvd 2491	. . . . . . . . . . . . . . . . . 18 |- (-. A.x x = y -> F/_yr)
34	32, 33	nfeqd 2504	. . . . . . . . . . . . . . . . 17 |- (-. A.x x = y -> F/y x = r)
35	34	exdistrf 1971	. . . . . . . . . . . . . . . 16 |- (E.xE.y(x = r /\ E.z(y = s /\ (z = t /\ ph))) -> E.x(x = r /\ E.yE.z(y = s /\ (z = t /\ ph))))
36		nfcvf2 2513	. . . . . . . . . . . . . . . . . . . 20 |- (-. A.y y = z -> F/_zy)
37		nfcvd 2491	. . . . . . . . . . . . . . . . . . . 20 |- (-. A.y y = z -> F/_zs)
38	36, 37	nfeqd 2504	. . . . . . . . . . . . . . . . . . 19 |- (-. A.y y = z -> F/z y = s)
39	38	exdistrf 1971	. . . . . . . . . . . . . . . . . 18 |- (E.yE.z(y = s /\ (z = t /\ ph)) -> E.y(y = s /\ E.z(z = t /\ ph)))
40	39	anim2i 552	. . . . . . . . . . . . . . . . 17 |- ((x = r /\ E.yE.z(y = s /\ (z = t /\ ph))) -> (x = r /\ E.y(y = s /\ E.z(z = t /\ ph))))
41	40	eximi 1576	. . . . . . . . . . . . . . . 16 |- (E.x(x = r /\ E.yE.z(y = s /\ (z = t /\ ph))) -> E.x(x = r /\ E.y(y = s /\ E.z(z = t /\ ph))))
42	31, 35, 41	3syl 18	. . . . . . . . . . . . . . 15 |- (E.xE.yE.z(x = r /\ (y = s /\ (z = t /\ ph))) -> E.x(x = r /\ E.y(y = s /\ E.z(z = t /\ ph))))
43	23, 42	sylbi 187	. . . . . . . . . . . . . 14 |- (E.xE.yE.z(<.<.x, y>., z>. = <.<.r, s>., t>. /\ ph) -> E.x(x = r /\ E.y(y = s /\ E.z(z = t /\ ph))))
44		df-3an 936	. . . . . . . . . . . . . . . . 17 |- ((x = r /\ y = s /\ z = t) <-> ((x = r /\ y = s) /\ z = t))
45	18, 44	bitr4i 243	. . . . . . . . . . . . . . . 16 |- (<.<.x, y>., z>. = <.<.r, s>., t>. <-> (x = r /\ y = s /\ z = t))
46		euequ1 2292	. . . . . . . . . . . . . . . . . . 19 |- E!x x = r
47		eupick 2267	. . . . . . . . . . . . . . . . . . 19 |- ((E!x x = r /\ E.x(x = r /\ E.y(y = s /\ E.z(z = t /\ ph)))) -> (x = r -> E.y(y = s /\ E.z(z = t /\ ph))))
48	46, 47	mpan 651	. . . . . . . . . . . . . . . . . 18 |- (E.x(x = r /\ E.y(y = s /\ E.z(z = t /\ ph))) -> (x = r -> E.y(y = s /\ E.z(z = t /\ ph))))
49		euequ1 2292	. . . . . . . . . . . . . . . . . . . 20 |- E!y y = s
50		eupick 2267	. . . . . . . . . . . . . . . . . . . 20 |- ((E!y y = s /\ E.y(y = s /\ E.z(z = t /\ ph))) -> (y = s -> E.z(z = t /\ ph)))
51	49, 50	mpan 651	. . . . . . . . . . . . . . . . . . 19 |- (E.y(y = s /\ E.z(z = t /\ ph)) -> (y = s -> E.z(z = t /\ ph)))
52		euequ1 2292	. . . . . . . . . . . . . . . . . . . 20 |- E!z z = t
53		eupick 2267	. . . . . . . . . . . . . . . . . . . 20 |- ((E!z z = t /\ E.z(z = t /\ ph)) -> (z = t -> ph))
54	52, 53	mpan 651	. . . . . . . . . . . . . . . . . . 19 |- (E.z(z = t /\ ph) -> (z = t -> ph))
55	51, 54	syl6 29	. . . . . . . . . . . . . . . . . 18 |- (E.y(y = s /\ E.z(z = t /\ ph)) -> (y = s -> (z = t -> ph)))
56	48, 55	syl6 29	. . . . . . . . . . . . . . . . 17 |- (E.x(x = r /\ E.y(y = s /\ E.z(z = t /\ ph))) -> (x = r -> (y = s -> (z = t -> ph))))
57	56	3impd 1165	. . . . . . . . . . . . . . . 16 |- (E.x(x = r /\ E.y(y = s /\ E.z(z = t /\ ph))) -> ((x = r /\ y = s /\ z = t) -> ph))
58	45, 57	syl5bi 208	. . . . . . . . . . . . . . 15 |- (E.x(x = r /\ E.y(y = s /\ E.z(z = t /\ ph))) -> (<.<.x, y>., z>. = <.<.r, s>., t>. -> ph))
59	58	com12 27	. . . . . . . . . . . . . 14 |- (<.<.x, y>., z>. = <.<.r, s>., t>. -> (E.x(x = r /\ E.y(y = s /\ E.z(z = t /\ ph))) -> ph))
60	43, 59	syl5 28	. . . . . . . . . . . . 13 |- (<.<.x, y>., z>. = <.<.r, s>., t>. -> (E.xE.yE.z(<.<.x, y>., z>. = <.<.r, s>., t>. /\ ph) -> ph))
61		eqeq1 2359	. . . . . . . . . . . . . . 15 |- (w = <.<.r, s>., t>. -> (w = <.<.x, y>., z>. <-> <.<.r, s>., t>. = <.<.x, y>., z>.))
62		eqcom 2355	. . . . . . . . . . . . . . 15 |- (<.<.r, s>., t>. = <.<.x, y>., z>. <-> <.<.x, y>., z>. = <.<.r, s>., t>.)
63	61, 62	syl6bb 252	. . . . . . . . . . . . . 14 |- (w = <.<.r, s>., t>. -> (w = <.<.x, y>., z>. <-> <.<.x, y>., z>. = <.<.r, s>., t>.))
64	63	anbi1d 685	. . . . . . . . . . . . . . . 16 |- (w = <.<.r, s>., t>. -> ((w = <.<.x, y>., z>. /\ ph) <-> (<.<.x, y>., z>. = <.<.r, s>., t>. /\ ph)))
65	64	3exbidv 1629	. . . . . . . . . . . . . . 15 |- (w = <.<.r, s>., t>. -> (E.xE.yE.z(w = <.<.x, y>., z>. /\ ph) <-> E.xE.yE.z(<.<.x, y>., z>. = <.<.r, s>., t>. /\ ph)))
66	65	imbi1d 308	. . . . . . . . . . . . . 14 |- (w = <.<.r, s>., t>. -> ((E.xE.yE.z(w = <.<.x, y>., z>. /\ ph) -> ph) <-> (E.xE.yE.z(<.<.x, y>., z>. = <.<.r, s>., t>. /\ ph) -> ph)))
67	63, 66	imbi12d 311	. . . . . . . . . . . . 13 |- (w = <.<.r, s>., t>. -> ((w = <.<.x, y>., z>. -> (E.xE.yE.z(w = <.<.x, y>., z>. /\ ph) -> ph)) <-> (<.<.x, y>., z>. = <.<.r, s>., t>. -> (E.xE.yE.z(<.<.x, y>., z>. = <.<.r, s>., t>. /\ ph) -> ph))))
68	60, 67	mpbiri 224	. . . . . . . . . . . 12 |- (w = <.<.r, s>., t>. -> (w = <.<.x, y>., z>. -> (E.xE.yE.z(w = <.<.x, y>., z>. /\ ph) -> ph)))
69	14, 68	syl6bi 219	. . . . . . . . . . 11 |- (a = <.r, s>. -> (w = <.a, t>. -> (w = <.<.x, y>., z>. -> (E.xE.yE.z(w = <.<.x, y>., z>. /\ ph) -> ph))))
70	69	adantr 451	. . . . . . . . . 10 |- ((a = <.r, s>. /\ <.r, s>. = <.x, y>.) -> (w = <.a, t>. -> (w = <.<.x, y>., z>. -> (E.xE.yE.z(w = <.<.x, y>., z>. /\ ph) -> ph))))
71	70	exlimivv 1635	. . . . . . . . 9 |- (E.rE.s(a = <.r, s>. /\ <.r, s>. = <.x, y>.) -> (w = <.a, t>. -> (w = <.<.x, y>., z>. -> (E.xE.yE.z(w = <.<.x, y>., z>. /\ ph) -> ph))))
72	12, 71	sylbi 187	. . . . . . . 8 |- (a = <.x, y>. -> (w = <.a, t>. -> (w = <.<.x, y>., z>. -> (E.xE.yE.z(w = <.<.x, y>., z>. /\ ph) -> ph))))
73	72	com3l 75	. . . . . . 7 |- (w = <.a, t>. -> (w = <.<.x, y>., z>. -> (a = <.x, y>. -> (E.xE.yE.z(w = <.<.x, y>., z>. /\ ph) -> ph))))
74	11, 73	mpdd 36	. . . . . 6 |- (w = <.a, t>. -> (w = <.<.x, y>., z>. -> (E.xE.yE.z(w = <.<.x, y>., z>. /\ ph) -> ph)))
75	74	adantr 451	. . . . 5 |- ((w = <.a, t>. /\ <.a, t>. = <.<.x, y>., z>.) -> (w = <.<.x, y>., z>. -> (E.xE.yE.z(w = <.<.x, y>., z>. /\ ph) -> ph)))
76	75	exlimivv 1635	. . . 4 |- (E.aE.t(w = <.a, t>. /\ <.a, t>. = <.<.x, y>., z>.) -> (w = <.<.x, y>., z>. -> (E.xE.yE.z(w = <.<.x, y>., z>. /\ ph) -> ph)))
77	7, 76	mpcom 32	. . 3 |- (w = <.<.x, y>., z>. -> (E.xE.yE.z(w = <.<.x, y>., z>. /\ ph) -> ph))
78		19.8a 1756	. . . . 5 |- ((w = <.<.x, y>., z>. /\ ph) -> E.z(w = <.<.x, y>., z>. /\ ph))
79		19.8a 1756	. . . . 5 |- (E.z(w = <.<.x, y>., z>. /\ ph) -> E.yE.z(w = <.<.x, y>., z>. /\ ph))
80		19.8a 1756	. . . . 5 |- (E.yE.z(w = <.<.x, y>., z>. /\ ph) -> E.xE.yE.z(w = <.<.x, y>., z>. /\ ph))
81	78, 79, 80	3syl 18	. . . 4 |- ((w = <.<.x, y>., z>. /\ ph) -> E.xE.yE.z(w = <.<.x, y>., z>. /\ ph))
82	81	ex 423	. . 3 |- (w = <.<.x, y>., z>. -> (ph -> E.xE.yE.z(w = <.<.x, y>., z>. /\ ph)))
83	77, 82	impbid 183	. 2 |- (w = <.<.x, y>., z>. -> (E.xE.yE.z(w = <.<.x, y>., z>. /\ ph) <-> ph))
84		df-oprab 5529	. 2 |- {<.<.x, y>., z>. | ph} = {w | E.xE.yE.z(w = <.<.x, y>., z>. /\ ph)}
85	5, 83, 84	elab2 2989	1 |- (<.<.x, y>., z>. e. {<.<.x, y>., z>. | ph} <-> ph)

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  E!weu 2204  <.cop 4562  {coprab 5528

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-oprab 5529

This theorem is referenced by:  ovidig  5594