Theorem eloprabga 5578

Description: The law of concretion for operation class abstraction. Compare elopab 4696. (Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy, 18-Jun-2012.) Removed unnecessary distinct variable requirements. (Revised by Mario Carneiro, 19-Dec-2013.)

Hypothesis

Ref	Expression
eloprabga.1	|- ((x = A /\ y = B /\ z = C) -> (ph <-> ps))

Assertion

Ref	Expression
eloprabga	|- ((A e. V /\ B e. W /\ C e. X) -> (<.<.A, B>., C>. e. {<.<.x, y>., z>. | ph} <-> ps))

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

Allowed substitution hints:   ph(x,y,z)   V(x,y,z)   W(x,y,z)   X(x,y,z)

Proof of Theorem eloprabga

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2867	. 2 |- (A e. V -> A e. _V)
2		elex 2867	. 2 |- (B e. W -> B e. _V)
3		elex 2867	. 2 |- (C e. X -> C e. _V)
4		opexg 4587	. . . . 5 |- ((A e. _V /\ B e. _V) -> <.A, B>. e. _V)
5		opexg 4587	. . . . 5 |- ((<.A, B>. e. _V /\ C e. _V) -> <.<.A, B>., C>. e. _V)
6	4, 5	sylan 457	. . . 4 |- (((A e. _V /\ B e. _V) /\ C e. _V) -> <.<.A, B>., C>. e. _V)
7	6	3impa 1146	. . 3 |- ((A e. _V /\ B e. _V /\ C e. _V) -> <.<.A, B>., C>. e. _V)
8		eqeq1 2359	. . . . . . . . . . 11 |- (w = <.<.A, B>., C>. -> (w = <.<.x, y>., z>. <-> <.<.A, B>., C>. = <.<.x, y>., z>.))
9		eqcom 2355	. . . . . . . . . . . 12 |- (<.<.A, B>., C>. = <.<.x, y>., z>. <-> <.<.x, y>., z>. = <.<.A, B>., C>.)
10		opth 4602	. . . . . . . . . . . . . 14 |- (<.x, y>. = <.A, B>. <-> (x = A /\ y = B))
11	10	anbi1i 676	. . . . . . . . . . . . 13 |- ((<.x, y>. = <.A, B>. /\ z = C) <-> ((x = A /\ y = B) /\ z = C))
12		opth 4602	. . . . . . . . . . . . 13 |- (<.<.x, y>., z>. = <.<.A, B>., C>. <-> (<.x, y>. = <.A, B>. /\ z = C))
13		df-3an 936	. . . . . . . . . . . . 13 |- ((x = A /\ y = B /\ z = C) <-> ((x = A /\ y = B) /\ z = C))
14	11, 12, 13	3bitr4i 268	. . . . . . . . . . . 12 |- (<.<.x, y>., z>. = <.<.A, B>., C>. <-> (x = A /\ y = B /\ z = C))
15	9, 14	bitri 240	. . . . . . . . . . 11 |- (<.<.A, B>., C>. = <.<.x, y>., z>. <-> (x = A /\ y = B /\ z = C))
16	8, 15	syl6bb 252	. . . . . . . . . 10 |- (w = <.<.A, B>., C>. -> (w = <.<.x, y>., z>. <-> (x = A /\ y = B /\ z = C)))
17	16	anbi1d 685	. . . . . . . . 9 |- (w = <.<.A, B>., C>. -> ((w = <.<.x, y>., z>. /\ ph) <-> ((x = A /\ y = B /\ z = C) /\ ph)))
18		eloprabga.1	. . . . . . . . . 10 |- ((x = A /\ y = B /\ z = C) -> (ph <-> ps))
19	18	pm5.32i 618	. . . . . . . . 9 |- (((x = A /\ y = B /\ z = C) /\ ph) <-> ((x = A /\ y = B /\ z = C) /\ ps))
20	17, 19	syl6bb 252	. . . . . . . 8 |- (w = <.<.A, B>., C>. -> ((w = <.<.x, y>., z>. /\ ph) <-> ((x = A /\ y = B /\ z = C) /\ ps)))
21	20	3exbidv 1629	. . . . . . 7 |- (w = <.<.A, B>., C>. -> (E.xE.yE.z(w = <.<.x, y>., z>. /\ ph) <-> E.xE.yE.z((x = A /\ y = B /\ z = C) /\ ps)))
22	21	adantl 452	. . . . . 6 |- (((A e. _V /\ B e. _V /\ C e. _V) /\ w = <.<.A, B>., C>.) -> (E.xE.yE.z(w = <.<.x, y>., z>. /\ ph) <-> E.xE.yE.z((x = A /\ y = B /\ z = C) /\ ps)))
23		df-oprab 5528	. . . . . . . . . 10 |- {<.<.x, y>., z>. | ph} = {w | E.xE.yE.z(w = <.<.x, y>., z>. /\ ph)}
24	23	eleq2i 2417	. . . . . . . . 9 |- (w e. {<.<.x, y>., z>. | ph} <-> w e. {w | E.xE.yE.z(w = <.<.x, y>., z>. /\ ph)})
25		abid 2341	. . . . . . . . 9 |- (w e. {w | E.xE.yE.z(w = <.<.x, y>., z>. /\ ph)} <-> E.xE.yE.z(w = <.<.x, y>., z>. /\ ph))
26	24, 25	bitr2i 241	. . . . . . . 8 |- (E.xE.yE.z(w = <.<.x, y>., z>. /\ ph) <-> w e. {<.<.x, y>., z>. | ph})
27		eleq1 2413	. . . . . . . 8 |- (w = <.<.A, B>., C>. -> (w e. {<.<.x, y>., z>. | ph} <-> <.<.A, B>., C>. e. {<.<.x, y>., z>. | ph}))
28	26, 27	syl5bb 248	. . . . . . 7 |- (w = <.<.A, B>., C>. -> (E.xE.yE.z(w = <.<.x, y>., z>. /\ ph) <-> <.<.A, B>., C>. e. {<.<.x, y>., z>. | ph}))
29	28	adantl 452	. . . . . 6 |- (((A e. _V /\ B e. _V /\ C e. _V) /\ w = <.<.A, B>., C>.) -> (E.xE.yE.z(w = <.<.x, y>., z>. /\ ph) <-> <.<.A, B>., C>. e. {<.<.x, y>., z>. | ph}))
30		isset 2863	. . . . . . . . . . . 12 |- (A e. _V <-> E.x x = A)
31		isset 2863	. . . . . . . . . . . 12 |- (B e. _V <-> E.y y = B)
32		isset 2863	. . . . . . . . . . . 12 |- (C e. _V <-> E.z z = C)
33	30, 31, 32	3anbi123i 1140	. . . . . . . . . . 11 |- ((A e. _V /\ B e. _V /\ C e. _V) <-> (E.x x = A /\ E.y y = B /\ E.z z = C))
34		eeeanv 1914	. . . . . . . . . . 11 |- (E.xE.yE.z(x = A /\ y = B /\ z = C) <-> (E.x x = A /\ E.y y = B /\ E.z z = C))
35	33, 34	bitr4i 243	. . . . . . . . . 10 |- ((A e. _V /\ B e. _V /\ C e. _V) <-> E.xE.yE.z(x = A /\ y = B /\ z = C))
36	35	biimpi 186	. . . . . . . . 9 |- ((A e. _V /\ B e. _V /\ C e. _V) -> E.xE.yE.z(x = A /\ y = B /\ z = C))
37	36	biantrurd 494	. . . . . . . 8 |- ((A e. _V /\ B e. _V /\ C e. _V) -> (ps <-> (E.xE.yE.z(x = A /\ y = B /\ z = C) /\ ps)))
38		19.41vvv 1903	. . . . . . . 8 |- (E.xE.yE.z((x = A /\ y = B /\ z = C) /\ ps) <-> (E.xE.yE.z(x = A /\ y = B /\ z = C) /\ ps))
39	37, 38	syl6rbbr 255	. . . . . . 7 |- ((A e. _V /\ B e. _V /\ C e. _V) -> (E.xE.yE.z((x = A /\ y = B /\ z = C) /\ ps) <-> ps))
40	39	adantr 451	. . . . . 6 |- (((A e. _V /\ B e. _V /\ C e. _V) /\ w = <.<.A, B>., C>.) -> (E.xE.yE.z((x = A /\ y = B /\ z = C) /\ ps) <-> ps))
41	22, 29, 40	3bitr3d 274	. . . . 5 |- (((A e. _V /\ B e. _V /\ C e. _V) /\ w = <.<.A, B>., C>.) -> (<.<.A, B>., C>. e. {<.<.x, y>., z>. | ph} <-> ps))
42	41	expcom 424	. . . 4 |- (w = <.<.A, B>., C>. -> ((A e. _V /\ B e. _V /\ C e. _V) -> (<.<.A, B>., C>. e. {<.<.x, y>., z>. | ph} <-> ps)))
43	42	vtocleg 2925	. . 3 |- (<.<.A, B>., C>. e. _V -> ((A e. _V /\ B e. _V /\ C e. _V) -> (<.<.A, B>., C>. e. {<.<.x, y>., z>. | ph} <-> ps)))
44	7, 43	mpcom 32	. 2 |- ((A e. _V /\ B e. _V /\ C e. _V) -> (<.<.A, B>., C>. e. {<.<.x, y>., z>. | ph} <-> ps))
45	1, 2, 3, 44	syl3an 1224	1 |- ((A e. V /\ B e. W /\ C e. X) -> (<.<.A, B>., C>. e. {<.<.x, y>., z>. | ph} <-> ps))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934  E.wex 1541   = wceq 1642   e. wcel 1710  {cab 2339  _Vcvv 2859  <.cop 4561  {coprab 5527

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-oprab 5528

This theorem is referenced by:  eloprabg  5579  ovigg  5596