Theorem eloprabi 6215

Description: A consequence of membership in an operation class abstraction, using ordered pair extractors. (Contributed by NM, 6-Nov-2006.) (Revised by David Abernethy, 19-Jun-2012.)

Hypotheses

Ref	Expression
eloprabi.1	|- ( x = ( 1st ` ( 1st ` A ) ) -> ( ph <-> ps ) )
eloprabi.2	|- ( y = ( 2nd ` ( 1st ` A ) ) -> ( ps <-> ch ) )
eloprabi.3	|- ( z = ( 2nd ` A ) -> ( ch <-> th ) )

Assertion

Ref	Expression
eloprabi	|- ( A e. { <. <. x , y >. , z >. | ph } -> th )

Distinct variable groups:   x, y, z, A   th, x, y, z

Allowed substitution hints:   ph( x, y, z)   ps( x, y, z)   ch( x, y, z)

Proof of Theorem eloprabi

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2196	. . . . . 6 |- ( w = A -> ( w = <. <. x , y >. , z >. <-> A = <. <. x , y >. , z >. ) )
2	1	anbi1d 465	. . . . 5 |- ( w = A -> ( ( w = <. <. x , y >. , z >. /\ ph ) <-> ( A = <. <. x , y >. , z >. /\ ph ) ) )
3	2	3exbidv 1880	. . . 4 |- ( w = A -> ( E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) <-> E. x E. y E. z ( A = <. <. x , y >. , z >. /\ ph ) ) )
4		df-oprab 5895	. . . 4 |- { <. <. x , y >. , z >. | ph } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) }
5	3, 4	elab2g 2899	. . 3 |- ( A e. { <. <. x , y >. , z >. | ph } -> ( A e. { <. <. x , y >. , z >. | ph } <-> E. x E. y E. z ( A = <. <. x , y >. , z >. /\ ph ) ) )
6	5	ibi 176	. 2 |- ( A e. { <. <. x , y >. , z >. | ph } -> E. x E. y E. z ( A = <. <. x , y >. , z >. /\ ph ) )
7		vex 2755	. . . . . . . . . . . 12 |- x e. _V
8		vex 2755	. . . . . . . . . . . 12 |- y e. _V
9	7, 8	opex 4244	. . . . . . . . . . 11 |- <. x , y >. e. _V
10		vex 2755	. . . . . . . . . . 11 |- z e. _V
11	9, 10	op1std 6167	. . . . . . . . . 10 |- ( A = <. <. x , y >. , z >. -> ( 1st ` A ) = <. x , y >. )
12	11	fveq2d 5534	. . . . . . . . 9 |- ( A = <. <. x , y >. , z >. -> ( 1st ` ( 1st ` A ) ) = ( 1st ` <. x , y >. ) )
13	7, 8	op1st 6165	. . . . . . . . 9 |- ( 1st ` <. x , y >. ) = x
14	12, 13	eqtr2di 2239	. . . . . . . 8 |- ( A = <. <. x , y >. , z >. -> x = ( 1st ` ( 1st ` A ) ) )
15		eloprabi.1	. . . . . . . 8 |- ( x = ( 1st ` ( 1st ` A ) ) -> ( ph <-> ps ) )
16	14, 15	syl 14	. . . . . . 7 |- ( A = <. <. x , y >. , z >. -> ( ph <-> ps ) )
17	11	fveq2d 5534	. . . . . . . . 9 |- ( A = <. <. x , y >. , z >. -> ( 2nd ` ( 1st ` A ) ) = ( 2nd ` <. x , y >. ) )
18	7, 8	op2nd 6166	. . . . . . . . 9 |- ( 2nd ` <. x , y >. ) = y
19	17, 18	eqtr2di 2239	. . . . . . . 8 |- ( A = <. <. x , y >. , z >. -> y = ( 2nd ` ( 1st ` A ) ) )
20		eloprabi.2	. . . . . . . 8 |- ( y = ( 2nd ` ( 1st ` A ) ) -> ( ps <-> ch ) )
21	19, 20	syl 14	. . . . . . 7 |- ( A = <. <. x , y >. , z >. -> ( ps <-> ch ) )
22	9, 10	op2ndd 6168	. . . . . . . . 9 |- ( A = <. <. x , y >. , z >. -> ( 2nd ` A ) = z )
23	22	eqcomd 2195	. . . . . . . 8 |- ( A = <. <. x , y >. , z >. -> z = ( 2nd ` A ) )
24		eloprabi.3	. . . . . . . 8 |- ( z = ( 2nd ` A ) -> ( ch <-> th ) )
25	23, 24	syl 14	. . . . . . 7 |- ( A = <. <. x , y >. , z >. -> ( ch <-> th ) )
26	16, 21, 25	3bitrd 214	. . . . . 6 |- ( A = <. <. x , y >. , z >. -> ( ph <-> th ) )
27	26	biimpa 296	. . . . 5 |- ( ( A = <. <. x , y >. , z >. /\ ph ) -> th )
28	27	exlimiv 1609	. . . 4 |- ( E. z ( A = <. <. x , y >. , z >. /\ ph ) -> th )
29	28	exlimiv 1609	. . 3 |- ( E. y E. z ( A = <. <. x , y >. , z >. /\ ph ) -> th )
30	29	exlimiv 1609	. 2 |- ( E. x E. y E. z ( A = <. <. x , y >. , z >. /\ ph ) -> th )
31	6, 30	syl 14	1 |- ( A e. { <. <. x , y >. , z >. | ph } -> th )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   E.wex 1503   e. wcel 2160   <.cop 3610   ` cfv 5231   {coprab 5892   1stc1st 6157   2ndc2nd 6158

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-iota 5193  df-fun 5233  df-fv 5239  df-oprab 5895  df-1st 6159  df-2nd 6160

This theorem is referenced by: (None)