Theorem eloprabg 4013

Description: The law of concretion for operation class abstraction. Compare elopab 2817.

Hypotheses

Ref	Expression
eloprabg.1	|- (x = A -> (ph <-> ps))
eloprabg.2	|- (y = B -> (ps <-> ch))
eloprabg.3	|- (z = C -> (ch <-> th))

Assertion

Ref	Expression
eloprabg	|- ((A e. D /\ B e. R /\ C e. S) -> (<.<.A, B>., C>. e. {<.<.x, y>., z>. | ph} <-> th))

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

Proof of Theorem eloprabg

Step	Hyp	Ref	Expression
1		opex 2788	. . 3 |- <.<.A, B>., C>. e. V
2		eqeq1 1484	. . . . . . . . . 10 |- (w = <.<.A, B>., C>. -> (w = <.<.x, y>., z>. <-> <.<.A, B>., C>. = <.<.x, y>., z>.))
3		eqcom 1480	. . . . . . . . . 10 |- (<.<.A, B>., C>. = <.<.x, y>., z>. <-> <.<.x, y>., z>. = <.<.A, B>., C>.)
4	2, 3	syl6bb 538	. . . . . . . . 9 |- (w = <.<.A, B>., C>. -> (w = <.<.x, y>., z>. <-> <.<.x, y>., z>. = <.<.A, B>., C>.))
5		visset 1816	. . . . . . . . . . 11 |- x e. V
6		visset 1816	. . . . . . . . . . 11 |- y e. V
7		visset 1816	. . . . . . . . . . 11 |- z e. V
8	5, 6, 7	otthg 2796	. . . . . . . . . 10 |- ((B e. V /\ C e. V) -> (<.<.x, y>., z>. = <.<.A, B>., C>. <-> (x = A /\ y = B /\ z = C)))
9	8	3adant1 799	. . . . . . . . 9 |- ((A e. V /\ B e. V /\ C e. V) -> (<.<.x, y>., z>. = <.<.A, B>., C>. <-> (x = A /\ y = B /\ z = C)))
10	4, 9	sylan9bbr 543	. . . . . . . 8 |- (((A e. V /\ B e. V /\ C e. V) /\ w = <.<.A, B>., C>.) -> (w = <.<.x, y>., z>. <-> (x = A /\ y = B /\ z = C)))
11	10	anbi1d 619	. . . . . . 7 |- (((A e. V /\ B e. V /\ C e. V) /\ w = <.<.A, B>., C>.) -> ((w = <.<.x, y>., z>. /\ ph) <-> ((x = A /\ y = B /\ z = C) /\ ph)))
12		eloprabg.1	. . . . . . . . 9 |- (x = A -> (ph <-> ps))
13		eloprabg.2	. . . . . . . . 9 |- (y = B -> (ps <-> ch))
14		eloprabg.3	. . . . . . . . 9 |- (z = C -> (ch <-> th))
15	12, 13, 14	syl3an9b 893	. . . . . . . 8 |- ((x = A /\ y = B /\ z = C) -> (ph <-> th))
16	15	pm5.32i 647	. . . . . . 7 |- (((x = A /\ y = B /\ z = C) /\ ph) <-> ((x = A /\ y = B /\ z = C) /\ th))
17	11, 16	syl6bb 538	. . . . . 6 |- (((A e. V /\ B e. V /\ C e. V) /\ w = <.<.A, B>., C>.) -> ((w = <.<.x, y>., z>. /\ ph) <-> ((x = A /\ y = B /\ z = C) /\ th)))
18	17	3exbidv 1284	. . . . 5 |- (((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) /\ th)))
19		eleq1 1537	. . . . . . 7 |- (w = <.<.A, B>., C>. -> (w e. {<.<.x, y>., z>. | ph} <-> <.<.A, B>., C>. e. {<.<.x, y>., z>. | ph}))
20		df-oprab 3972	. . . . . . . . 9 |- {<.<.x, y>., z>. | ph} = {w | E.xE.yE.z(w = <.<.x, y>., z>. /\ ph)}
21	20	eleq2i 1541	. . . . . . . 8 |- (w e. {<.<.x, y>., z>. | ph} <-> w e. {w | E.xE.yE.z(w = <.<.x, y>., z>. /\ ph)})
22		abid 1468	. . . . . . . 8 |- (w e. {w | E.xE.yE.z(w = <.<.x, y>., z>. /\ ph)} <-> E.xE.yE.z(w = <.<.x, y>., z>. /\ ph))
23	21, 22	bitr2 174	. . . . . . 7 |- (E.xE.yE.z(w = <.<.x, y>., z>. /\ ph) <-> w e. {<.<.x, y>., z>. | ph})
24	19, 23	syl5bb 534	. . . . . 6 |- (w = <.<.A, B>., C>. -> (E.xE.yE.z(w = <.<.x, y>., z>. /\ ph) <-> <.<.A, B>., C>. e. {<.<.x, y>., z>. | ph}))
25	24	adantl 390	. . . . 5 |- (((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}))
26		elex 1822	. . . . . . . . . 10 |- (A e. V -> E.x x = A)
27		elex 1822	. . . . . . . . . 10 |- (B e. V -> E.y y = B)
28		elex 1822	. . . . . . . . . 10 |- (C e. V -> E.z z = C)
29	26, 27, 28	3anim123i 823	. . . . . . . . 9 |- ((A e. V /\ B e. V /\ C e. V) -> (E.x x = A /\ E.y y = B /\ E.z z = C))
30		eeeanv 1326	. . . . . . . . 9 |- (E.xE.yE.z(x = A /\ y = B /\ z = C) <-> (E.x x = A /\ E.y y = B /\ E.z z = C))
31	29, 30	sylibr 200	. . . . . . . 8 |- ((A e. V /\ B e. V /\ C e. V) -> E.xE.yE.z(x = A /\ y = B /\ z = C))
32	31	biantrurd 729	. . . . . . 7 |- ((A e. V /\ B e. V /\ C e. V) -> (th <-> (E.xE.yE.z(x = A /\ y = B /\ z = C) /\ th)))
33		19.41vvv 1309	. . . . . . 7 |- (E.xE.yE.z((x = A /\ y = B /\ z = C) /\ th) <-> (E.xE.yE.z(x = A /\ y = B /\ z = C) /\ th))
34	32, 33	syl6rbbr 541	. . . . . 6 |- ((A e. V /\ B e. V /\ C e. V) -> (E.xE.yE.z((x = A /\ y = B /\ z = C) /\ th) <-> th))
35	34	adantr 391	. . . . 5 |- (((A e. V /\ B e. V /\ C e. V) /\ w = <.<.A, B>., C>.) -> (E.xE.yE.z((x = A /\ y = B /\ z = C) /\ th) <-> th))
36	18, 25, 35	3bitr3d 550	. . . 4 |- (((A e. V /\ B e. V /\ C e. V) /\ w = <.<.A, B>., C>.) -> (<.<.A, B>., C>. e. {<.<.x, y>., z>. | ph} <-> th))
37	36	expcom 374	. . 3 |- (w = <.<.A, B>., C>. -> ((A e. V /\ B e. V /\ C e. V) -> (<.<.A, B>., C>. e. {<.<.x, y>., z>. | ph} <-> th)))
38	1, 37	vtocle 1861	. 2 |- ((A e. V /\ B e. V /\ C e. V) -> (<.<.A, B>., C>. e. {<.<.x, y>., z>. | ph} <-> th))
39		elisset 1820	. 2 |- (A e. D -> A e. V)
40		elisset 1820	. 2 |- (B e. R -> B e. V)
41		elisset 1820	. 2 |- (C e. S -> C e. V)
42	38, 39, 40, 41	syl3an 870	1 |- ((A e. D /\ B e. R /\ C e. S) -> (<.<.A, B>., C>. e. {<.<.x, y>., z>. | ph} <-> th))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  E.wex 982  {cab 1466  Vcvv 1814  <.cop 2415  {copab2 3970

This theorem is referenced by:  oprabval 4029  oprabvalig 4030  eloprabi 4124  isnvlem 8225  isphg 8472  oprabvaligg 10435  ismgra 10613

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-oprab 3972

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