Theorem eloprabg 3035

Description: The law of concretion for operation class abstraction. Compare elopab 2110.

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 group(s):   x,y,z,A   x,B,y,z   x,C,y,z   x,D,y,z   x,R,y,z   x,S,y,z   ps,x   ch,x,y   th,x,y,z

Proof of Theorem eloprabg

Step	Hyp	Ref	Expression
1		opex 1893	. 2 |- <.<.A, B>., C>. e. V
2		cleq1 1107	. . . . . . . . . 10 |- (w = <.<.A, B>., C>. -> (w = <.<.x, y>., z>. <-> <.<.A, B>., C>. = <.<.x, y>., z>.))
3		cleqcom 1103	. . . . . . . . . 10 |- (<.<.A, B>., C>. = <.<.x, y>., z>. <-> <.<.x, y>., z>. = <.<.A, B>., C>.)
4	2, 3	syl6bb 414	. . . . . . . . 9 |- (w = <.<.A, B>., C>. -> (w = <.<.x, y>., z>. <-> <.<.x, y>., z>. = <.<.A, B>., C>.))
5		visset 1350	. . . . . . . . . . 11 |- x e. V
6		visset 1350	. . . . . . . . . . 11 |- y e. V
7		visset 1350	. . . . . . . . . . 11 |- z e. V
8	5, 6, 7	otthg 1900	. . . . . . . . . 10 |- ((B e. R /\ C e. S) -> (<.<.x, y>., z>. = <.<.A, B>., C>. <-> (x = A /\ y = B /\ z = C)))
9	8	3adant1 597	. . . . . . . . 9 |- ((A e. D /\ B e. R /\ C e. S) -> (<.<.x, y>., z>. = <.<.A, B>., C>. <-> (x = A /\ y = B /\ z = C)))
10	4, 9	sylan9bbr 419	. . . . . . . 8 |- (((A e. D /\ B e. R /\ C e. S) /\ w = <.<.A, B>., C>.) -> (w = <.<.x, y>., z>. <-> (x = A /\ y = B /\ z = C)))
11	10	anbi1d 469	. . . . . . 7 |- (((A e. D /\ B e. R /\ C e. S) /\ 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 634	. . . . . . . 8 |- ((x = A /\ y = B /\ z = C) -> (ph <-> th))
16	15	pm5.32i 489	. . . . . . 7 |- (((x = A /\ y = B /\ z = C) /\ ph) <-> ((x = A /\ y = B /\ z = C) /\ th))
17	11, 16	syl6bb 414	. . . . . 6 |- (((A e. D /\ B e. R /\ C e. S) /\ w = <.<.A, B>., C>.) -> ((w = <.<.x, y>., z>. /\ ph) <-> ((x = A /\ y = B /\ z = C) /\ th)))
18	17	bi3exdv 939	. . . . 5 |- (((A e. D /\ B e. R /\ C e. S) /\ 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 1149	. . . . . . 7 |- (w = <.<.A, B>., C>. -> (w e. {<.<.x, y>., z>. | ph} <-> <.<.A, B>., C>. e. {<.<.x, y>., z>. | ph}))
20		df-oprab 3004	. . . . . . . . 9 |- {<.<.x, y>., z>. | ph} = {w | E.xE.yE.z(w = <.<.x, y>., z>. /\ ph)}
21	20	eleq2i 1153	. . . . . . . 8 |- (w e. {<.<.x, y>., z>. | ph} <-> w e. {w | E.xE.yE.z(w = <.<.x, y>., z>. /\ ph)})
22		abid 1094	. . . . . . . 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 152	. . . . . . 7 |- (E.xE.yE.z(w = <.<.x, y>., z>. /\ ph) <-> w e. {<.<.x, y>., z>. | ph})
24	19, 23	syl5bb 410	. . . . . 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 305	. . . . 5 |- (((A e. D /\ B e. R /\ C e. S) /\ w = <.<.A, B>., C>.) -> (E.xE.yE.z(w = <.<.x, y>., z>. /\ ph) <-> <.<.A, B>., C>. e. {<.<.x, y>., z>. | ph}))
26		elex 1356	. . . . . . . . . 10 |- (A e. D -> E.x x = A)
27		elex 1356	. . . . . . . . . 10 |- (B e. R -> E.y y = B)
28		elex 1356	. . . . . . . . . 10 |- (C e. S -> E.z z = C)
29	26, 27, 28	im3an 605	. . . . . . . . 9 |- ((A e. D /\ B e. R /\ C e. S) -> (E.x x = A /\ E.y y = B /\ E.z z = C))
30		eeeanv 981	. . . . . . . . 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 175	. . . . . . . 8 |- ((A e. D /\ B e. R /\ C e. S) -> E.xE.yE.z(x = A /\ y = B /\ z = C))
32	31	biantrurd 546	. . . . . . 7 |- ((A e. D /\ B e. R /\ C e. S) -> (th <-> (E.xE.yE.z(x = A /\ y = B /\ z = C) /\ th)))
33		19.41vvv 965	. . . . . . 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 417	. . . . . 6 |- ((A e. D /\ B e. R /\ C e. S) -> (E.xE.yE.z((x = A /\ y = B /\ z = C) /\ th) <-> th))
35	34	adantr 306	. . . . 5 |- (((A e. D /\ B e. R /\ C e. S) /\ w = <.<.A, B>., C>.) -> (E.xE.yE.z((x = A /\ y = B /\ z = C) /\ th) <-> th))
36	18, 25, 35	3bitr3d 423	. . . 4 |- (((A e. D /\ B e. R /\ C e. S) /\ w = <.<.A, B>., C>.) -> (<.<.A, B>., C>. e. {<.<.x, y>., z>. | ph} <-> th))
37	36	exp 291	. . 3 |- ((A e. D /\ B e. R /\ C e. S) -> (w = <.<.A, B>., C>. -> (<.<.A, B>., C>. e. {<.<.x, y>., z>. | ph} <-> th)))
38	37	com12 13	. 2 |- (w = <.<.A, B>., C>. -> ((A e. D /\ B e. R /\ C e. S) -> (<.<.A, B>., C>. e. {<.<.x, y>., z>. | ph} <-> th)))
39	1, 38	vtocle 1391	1 |- ((A e. D /\ B e. R /\ C e. S) -> (<.<.A, B>., C>. e. {<.<.x, y>., z>. | ph} <-> th))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   /\ w3a 581  E.wex 678  {cab 1090   = wceq 1091   e. wcel 1092  <.cop 1810  {copab2 3002

This theorem is referenced by:  oprabval 3047  oprabvalig 3048

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-oprab 3004

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