Theorem eloprabga 5858

Description: The law of concretion for operation class abstraction. Compare elopab 4180. (Contributed by NM, 14-Sep-1999.) (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (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 2697	. 2 |- ( A e. V -> A e. _V )
2		elex 2697	. 2 |- ( B e. W -> B e. _V )
3		elex 2697	. 2 |- ( C e. X -> C e. _V )
4		opexg 4150	. . . . 5 |- ( ( A e. _V /\ B e. _V ) -> <. A , B >. e. _V )
5		opexg 4150	. . . . 5 |- ( ( <. A , B >. e. _V /\ C e. _V ) -> <. <. A , B >. , C >. e. _V )
6	4, 5	sylan 281	. . . 4 |- ( ( ( A e. _V /\ B e. _V ) /\ C e. _V ) -> <. <. A , B >. , C >. e. _V )
7	6	3impa 1176	. . 3 |- ( ( A e. _V /\ B e. _V /\ C e. _V ) -> <. <. A , B >. , C >. e. _V )
8		simpr 109	. . . . . . . . . . 11 |- ( ( ( A e. _V /\ B e. _V /\ C e. _V ) /\ w = <. <. A , B >. , C >. ) -> w = <. <. A , B >. , C >. )
9	8	eqeq1d 2148	. . . . . . . . . 10 |- ( ( ( A e. _V /\ B e. _V /\ C e. _V ) /\ w = <. <. A , B >. , C >. ) -> ( w = <. <. x , y >. , z >. <-> <. <. A , B >. , C >. = <. <. x , y >. , z >. ) )
10		eqcom 2141	. . . . . . . . . . 11 |- ( <. <. A , B >. , C >. = <. <. x , y >. , z >. <-> <. <. x , y >. , z >. = <. <. A , B >. , C >. )
11		vex 2689	. . . . . . . . . . . 12 |- x e. _V
12		vex 2689	. . . . . . . . . . . 12 |- y e. _V
13		vex 2689	. . . . . . . . . . . 12 |- z e. _V
14	11, 12, 13	otth2 4163	. . . . . . . . . . 11 |- ( <. <. x , y >. , z >. = <. <. A , B >. , C >. <-> ( x = A /\ y = B /\ z = C ) )
15	10, 14	bitri 183	. . . . . . . . . 10 |- ( <. <. A , B >. , C >. = <. <. x , y >. , z >. <-> ( x = A /\ y = B /\ z = C ) )
16	9, 15	syl6bb 195	. . . . . . . . 9 |- ( ( ( A e. _V /\ B e. _V /\ C e. _V ) /\ w = <. <. A , B >. , C >. ) -> ( w = <. <. x , y >. , z >. <-> ( x = A /\ y = B /\ z = C ) ) )
17	16	anbi1d 460	. . . . . . . 8 |- ( ( ( 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 ) ) )
18		eloprabga.1	. . . . . . . . 9 |- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) )
19	18	pm5.32i 449	. . . . . . . 8 |- ( ( ( x = A /\ y = B /\ z = C ) /\ ph ) <-> ( ( x = A /\ y = B /\ z = C ) /\ ps ) )
20	17, 19	syl6bb 195	. . . . . . 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 ) /\ ps ) ) )
21	20	3exbidv 1841	. . . . . 6 |- ( ( ( A e. _V /\ B e. _V /\ C e. _V ) /\ w = <. <. A , B >. , C >. ) -> ( E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) <-> E. x E. y E. z ( ( x = A /\ y = B /\ z = C ) /\ ps ) ) )
22		df-oprab 5778	. . . . . . . . . 10 |- { <. <. x , y >. , z >. | ph } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) }
23	22	eleq2i 2206	. . . . . . . . 9 |- ( w e. { <. <. x , y >. , z >. | ph } <-> w e. { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) } )
24		abid 2127	. . . . . . . . 9 |- ( w e. { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) } <-> E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) )
25	23, 24	bitr2i 184	. . . . . . . 8 |- ( E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) <-> w e. { <. <. x , y >. , z >. | ph } )
26		eleq1 2202	. . . . . . . 8 |- ( w = <. <. A , B >. , C >. -> ( w e. { <. <. x , y >. , z >. | ph } <-> <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } ) )
27	25, 26	syl5bb 191	. . . . . . 7 |- ( w = <. <. A , B >. , C >. -> ( E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) <-> <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } ) )
28	27	adantl 275	. . . . . 6 |- ( ( ( A e. _V /\ B e. _V /\ C e. _V ) /\ w = <. <. A , B >. , C >. ) -> ( E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) <-> <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } ) )
29		elisset 2700	. . . . . . . . . . 11 |- ( A e. _V -> E. x x = A )
30		elisset 2700	. . . . . . . . . . 11 |- ( B e. _V -> E. y y = B )
31		elisset 2700	. . . . . . . . . . 11 |- ( C e. _V -> E. z z = C )
32	29, 30, 31	3anim123i 1166	. . . . . . . . . 10 |- ( ( A e. _V /\ B e. _V /\ C e. _V ) -> ( E. x x = A /\ E. y y = B /\ E. z z = C ) )
33		eeeanv 1905	. . . . . . . . . 10 |- ( E. x E. y E. z ( x = A /\ y = B /\ z = C ) <-> ( E. x x = A /\ E. y y = B /\ E. z z = C ) )
34	32, 33	sylibr 133	. . . . . . . . 9 |- ( ( A e. _V /\ B e. _V /\ C e. _V ) -> E. x E. y E. z ( x = A /\ y = B /\ z = C ) )
35	34	biantrurd 303	. . . . . . . 8 |- ( ( A e. _V /\ B e. _V /\ C e. _V ) -> ( ps <-> ( E. x E. y E. z ( x = A /\ y = B /\ z = C ) /\ ps ) ) )
36		19.41vvv 1876	. . . . . . . 8 |- ( E. x E. y E. z ( ( x = A /\ y = B /\ z = C ) /\ ps ) <-> ( E. x E. y E. z ( x = A /\ y = B /\ z = C ) /\ ps ) )
37	35, 36	syl6rbbr 198	. . . . . . 7 |- ( ( A e. _V /\ B e. _V /\ C e. _V ) -> ( E. x E. y E. z ( ( x = A /\ y = B /\ z = C ) /\ ps ) <-> ps ) )
38	37	adantr 274	. . . . . 6 |- ( ( ( A e. _V /\ B e. _V /\ C e. _V ) /\ w = <. <. A , B >. , C >. ) -> ( E. x E. y E. z ( ( x = A /\ y = B /\ z = C ) /\ ps ) <-> ps ) )
39	21, 28, 38	3bitr3d 217	. . . . 5 |- ( ( ( A e. _V /\ B e. _V /\ C e. _V ) /\ w = <. <. A , B >. , C >. ) -> ( <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } <-> ps ) )
40	39	expcom 115	. . . 4 |- ( w = <. <. A , B >. , C >. -> ( ( A e. _V /\ B e. _V /\ C e. _V ) -> ( <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } <-> ps ) ) )
41	40	vtocleg 2757	. . 3 |- ( <. <. A , B >. , C >. e. _V -> ( ( A e. _V /\ B e. _V /\ C e. _V ) -> ( <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } <-> ps ) ) )
42	7, 41	mpcom 36	. 2 |- ( ( A e. _V /\ B e. _V /\ C e. _V ) -> ( <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } <-> ps ) )
43	1, 2, 3, 42	syl3an 1258	1 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   E.wex 1468   e. wcel 1480   {cab 2125   _Vcvv 2686   <.cop 3530   {coprab 5775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-oprab 5778

This theorem is referenced by:  eloprabg  5859  ovigg  5891