Theorem eloprabga 6747

Description: The law of concretion for operation class abstraction. Compare elopab 4983. (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 3212	. 2 |- ( A e. V -> A e. _V )
2		elex 3212	. 2 |- ( B e. W -> B e. _V )
3		elex 3212	. 2 |- ( C e. X -> C e. _V )
4		opex 4932	. . 3 |- <. <. A , B >. , C >. e. _V
5		simpr 477	. . . . . . . . . 10 |- ( ( ( A e. _V /\ B e. _V /\ C e. _V ) /\ w = <. <. A , B >. , C >. ) -> w = <. <. A , B >. , C >. )
6	5	eqeq1d 2624	. . . . . . . . 9 |- ( ( ( A e. _V /\ B e. _V /\ C e. _V ) /\ w = <. <. A , B >. , C >. ) -> ( w = <. <. x , y >. , z >. <-> <. <. A , B >. , C >. = <. <. x , y >. , z >. ) )
7		eqcom 2629	. . . . . . . . . 10 |- ( <. <. A , B >. , C >. = <. <. x , y >. , z >. <-> <. <. x , y >. , z >. = <. <. A , B >. , C >. )
8		vex 3203	. . . . . . . . . . 11 |- x e. _V
9		vex 3203	. . . . . . . . . . 11 |- y e. _V
10		vex 3203	. . . . . . . . . . 11 |- z e. _V
11	8, 9, 10	otth2 4952	. . . . . . . . . 10 |- ( <. <. x , y >. , z >. = <. <. A , B >. , C >. <-> ( x = A /\ y = B /\ z = C ) )
12	7, 11	bitri 264	. . . . . . . . 9 |- ( <. <. A , B >. , C >. = <. <. x , y >. , z >. <-> ( x = A /\ y = B /\ z = C ) )
13	6, 12	syl6bb 276	. . . . . . . 8 |- ( ( ( A e. _V /\ B e. _V /\ C e. _V ) /\ w = <. <. A , B >. , C >. ) -> ( w = <. <. x , y >. , z >. <-> ( x = A /\ y = B /\ z = C ) ) )
14	13	anbi1d 741	. . . . . . 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 ) ) )
15		eloprabga.1	. . . . . . . 8 |- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) )
16	15	pm5.32i 669	. . . . . . 7 |- ( ( ( x = A /\ y = B /\ z = C ) /\ ph ) <-> ( ( x = A /\ y = B /\ z = C ) /\ ps ) )
17	14, 16	syl6bb 276	. . . . . 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 ) /\ ps ) ) )
18	17	3exbidv 1853	. . . . 5 |- ( ( ( 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 ) ) )
19		df-oprab 6654	. . . . . . . . 9 |- { <. <. x , y >. , z >. | ph } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) }
20	19	eleq2i 2693	. . . . . . . 8 |- ( w e. { <. <. x , y >. , z >. | ph } <-> w e. { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) } )
21		abid 2610	. . . . . . . 8 |- ( 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 ) )
22	20, 21	bitr2i 265	. . . . . . 7 |- ( E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) <-> w e. { <. <. x , y >. , z >. | ph } )
23		eleq1 2689	. . . . . . 7 |- ( w = <. <. A , B >. , C >. -> ( w e. { <. <. x , y >. , z >. | ph } <-> <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } ) )
24	22, 23	syl5bb 272	. . . . . 6 |- ( w = <. <. A , B >. , C >. -> ( E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) <-> <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } ) )
25	24	adantl 482	. . . . 5 |- ( ( ( 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 } ) )
26		elisset 3215	. . . . . . . . . 10 |- ( A e. _V -> E. x x = A )
27		elisset 3215	. . . . . . . . . 10 |- ( B e. _V -> E. y y = B )
28		elisset 3215	. . . . . . . . . 10 |- ( C e. _V -> E. z z = C )
29	26, 27, 28	3anim123i 1247	. . . . . . . . 9 |- ( ( A e. _V /\ B e. _V /\ C e. _V ) -> ( E. x x = A /\ E. y y = B /\ E. z z = C ) )
30		eeeanv 2183	. . . . . . . . 9 |- ( E. x E. y E. z ( x = A /\ y = B /\ z = C ) <-> ( E. x x = A /\ E. y y = B /\ E. z z = C ) )
31	29, 30	sylibr 224	. . . . . . . 8 |- ( ( A e. _V /\ B e. _V /\ C e. _V ) -> E. x E. y E. z ( x = A /\ y = B /\ z = C ) )
32	31	biantrurd 529	. . . . . . 7 |- ( ( A e. _V /\ B e. _V /\ C e. _V ) -> ( ps <-> ( E. x E. y E. z ( x = A /\ y = B /\ z = C ) /\ ps ) ) )
33		19.41vvv 1916	. . . . . . 7 |- ( 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 ) )
34	32, 33	syl6rbbr 279	. . . . . 6 |- ( ( A e. _V /\ B e. _V /\ C e. _V ) -> ( E. x E. y E. z ( ( x = A /\ y = B /\ z = C ) /\ ps ) <-> ps ) )
35	34	adantr 481	. . . . 5 |- ( ( ( 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 ) )
36	18, 25, 35	3bitr3d 298	. . . 4 |- ( ( ( A e. _V /\ B e. _V /\ C e. _V ) /\ w = <. <. A , B >. , C >. ) -> ( <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } <-> ps ) )
37	36	expcom 451	. . 3 |- ( w = <. <. A , B >. , C >. -> ( ( A e. _V /\ B e. _V /\ C e. _V ) -> ( <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } <-> ps ) ) )
38	4, 37	vtocle 3282	. 2 |- ( ( A e. _V /\ B e. _V /\ C e. _V ) -> ( <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } <-> ps ) )
39	1, 2, 3, 38	syl3an 1368	1 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   _Vcvv 3200   <.cop 4183   {coprab 6651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-oprab 6654

This theorem is referenced by:  eloprabg  6748  ovigg  6781  vdwpc  15684  elmpps  31470  uncov  33390