Theorem eloprabga 6107

Description: The law of concretion for operation class abstraction. Compare elopab 4352. (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 2814	. 2 |- ( A e. V -> A e. _V )
2		elex 2814	. 2 |- ( B e. W -> B e. _V )
3		elex 2814	. 2 |- ( C e. X -> C e. _V )
4		opexg 4320	. . . . 5 |- ( ( A e. _V /\ B e. _V ) -> <. A , B >. e. _V )
5		opexg 4320	. . . . 5 |- ( ( <. A , B >. e. _V /\ C e. _V ) -> <. <. A , B >. , C >. e. _V )
6	4, 5	sylan 283	. . . 4 |- ( ( ( A e. _V /\ B e. _V ) /\ C e. _V ) -> <. <. A , B >. , C >. e. _V )
7	6	3impa 1220	. . 3 |- ( ( A e. _V /\ B e. _V /\ C e. _V ) -> <. <. A , B >. , C >. e. _V )
8		simpr 110	. . . . . . . . . . 11 |- ( ( ( A e. _V /\ B e. _V /\ C e. _V ) /\ w = <. <. A , B >. , C >. ) -> w = <. <. A , B >. , C >. )
9	8	eqeq1d 2240	. . . . . . . . . 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 2233	. . . . . . . . . . 11 |- ( <. <. A , B >. , C >. = <. <. x , y >. , z >. <-> <. <. x , y >. , z >. = <. <. A , B >. , C >. )
11		vex 2805	. . . . . . . . . . . 12 |- x e. _V
12		vex 2805	. . . . . . . . . . . 12 |- y e. _V
13		vex 2805	. . . . . . . . . . . 12 |- z e. _V
14	11, 12, 13	otth2 4333	. . . . . . . . . . 11 |- ( <. <. x , y >. , z >. = <. <. A , B >. , C >. <-> ( x = A /\ y = B /\ z = C ) )
15	10, 14	bitri 184	. . . . . . . . . 10 |- ( <. <. A , B >. , C >. = <. <. x , y >. , z >. <-> ( x = A /\ y = B /\ z = C ) )
16	9, 15	bitrdi 196	. . . . . . . . 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 465	. . . . . . . 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 454	. . . . . . . 8 |- ( ( ( x = A /\ y = B /\ z = C ) /\ ph ) <-> ( ( x = A /\ y = B /\ z = C ) /\ ps ) )
20	17, 19	bitrdi 196	. . . . . . 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 1917	. . . . . 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 6021	. . . . . . . . . 10 |- { <. <. x , y >. , z >. | ph } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) }
23	22	eleq2i 2298	. . . . . . . . 9 |- ( w e. { <. <. x , y >. , z >. | ph } <-> w e. { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) } )
24		abid 2219	. . . . . . . . 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 185	. . . . . . . 8 |- ( E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) <-> w e. { <. <. x , y >. , z >. | ph } )
26		eleq1 2294	. . . . . . . 8 |- ( w = <. <. A , B >. , C >. -> ( w e. { <. <. x , y >. , z >. | ph } <-> <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } ) )
27	25, 26	bitrid 192	. . . . . . 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 277	. . . . . 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		19.41vvv 1953	. . . . . . . 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 ) )
30		elisset 2817	. . . . . . . . . . 11 |- ( A e. _V -> E. x x = A )
31		elisset 2817	. . . . . . . . . . 11 |- ( B e. _V -> E. y y = B )
32		elisset 2817	. . . . . . . . . . 11 |- ( C e. _V -> E. z z = C )
33	30, 31, 32	3anim123i 1210	. . . . . . . . . 10 |- ( ( A e. _V /\ B e. _V /\ C e. _V ) -> ( E. x x = A /\ E. y y = B /\ E. z z = C ) )
34		eeeanv 1986	. . . . . . . . . 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 ) )
35	33, 34	sylibr 134	. . . . . . . . 9 |- ( ( A e. _V /\ B e. _V /\ C e. _V ) -> E. x E. y E. z ( x = A /\ y = B /\ z = C ) )
36	35	biantrurd 305	. . . . . . . 8 |- ( ( A e. _V /\ B e. _V /\ C e. _V ) -> ( ps <-> ( E. x E. y E. z ( x = A /\ y = B /\ z = C ) /\ ps ) ) )
37	29, 36	bitr4id 199	. . . . . . 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 276	. . . . . 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 218	. . . . 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 116	. . . 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 2877	. . 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 1315	1 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   E.wex 1540   e. wcel 2202   {cab 2217   _Vcvv 2802   <.cop 3672   {coprab 6018

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-oprab 6021

This theorem is referenced by:  eloprabg  6108  ovigg  6141