Theorem eloprabi 6342

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 2236	. . . . . 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 1915	. . . 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 6005	. . . 4 |- { <. <. x , y >. , z >. | ph } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) }
5	3, 4	elab2g 2950	. . 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 2802	. . . . . . . . . . . 12 |- x e. _V
8		vex 2802	. . . . . . . . . . . 12 |- y e. _V
9	7, 8	opex 4315	. . . . . . . . . . 11 |- <. x , y >. e. _V
10		vex 2802	. . . . . . . . . . 11 |- z e. _V
11	9, 10	op1std 6294	. . . . . . . . . 10 |- ( A = <. <. x , y >. , z >. -> ( 1st ` A ) = <. x , y >. )
12	11	fveq2d 5631	. . . . . . . . 9 |- ( A = <. <. x , y >. , z >. -> ( 1st ` ( 1st ` A ) ) = ( 1st ` <. x , y >. ) )
13	7, 8	op1st 6292	. . . . . . . . 9 |- ( 1st ` <. x , y >. ) = x
14	12, 13	eqtr2di 2279	. . . . . . . 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 5631	. . . . . . . . 9 |- ( A = <. <. x , y >. , z >. -> ( 2nd ` ( 1st ` A ) ) = ( 2nd ` <. x , y >. ) )
18	7, 8	op2nd 6293	. . . . . . . . 9 |- ( 2nd ` <. x , y >. ) = y
19	17, 18	eqtr2di 2279	. . . . . . . 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 6295	. . . . . . . . 9 |- ( A = <. <. x , y >. , z >. -> ( 2nd ` A ) = z )
23	22	eqcomd 2235	. . . . . . . 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 1644	. . . 4 |- ( E. z ( A = <. <. x , y >. , z >. /\ ph ) -> th )
29	28	exlimiv 1644	. . 3 |- ( E. y E. z ( A = <. <. x , y >. , z >. /\ ph ) -> th )
30	29	exlimiv 1644	. 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 1395   E.wex 1538   e. wcel 2200   <.cop 3669   ` cfv 5318   {coprab 6002   1stc1st 6284   2ndc2nd 6285

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fv 5326  df-oprab 6005  df-1st 6286  df-2nd 6287

This theorem is referenced by: (None)