Theorem eloprabi 5822

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 2046	. . . . . 6 |- ( w = A -> ( w = <. <. x , y >. , z >. <-> A = <. <. x , y >. , z >. ) )
2	1	anbi1d 438	. . . . 5 |- ( w = A -> ( ( w = <. <. x , y >. , z >. /\ ph ) <-> ( A = <. <. x , y >. , z >. /\ ph ) ) )
3	2	3exbidv 1749	. . . 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 5516	. . . 4 |- { <. <. x , y >. , z >. | ph } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) }
5	3, 4	elab2g 2689	. . 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 165	. 2 |- ( A e. { <. <. x , y >. , z >. | ph } -> E. x E. y E. z ( A = <. <. x , y >. , z >. /\ ph ) )
7		vex 2560	. . . . . . . . . . . 12 |- x e. _V
8		vex 2560	. . . . . . . . . . . 12 |- y e. _V
9	7, 8	opex 3966	. . . . . . . . . . 11 |- <. x , y >. e. _V
10		vex 2560	. . . . . . . . . . 11 |- z e. _V
11	9, 10	op1std 5775	. . . . . . . . . 10 |- ( A = <. <. x , y >. , z >. -> ( 1st ` A ) = <. x , y >. )
12	11	fveq2d 5182	. . . . . . . . 9 |- ( A = <. <. x , y >. , z >. -> ( 1st ` ( 1st ` A ) ) = ( 1st ` <. x , y >. ) )
13	7, 8	op1st 5773	. . . . . . . . 9 |- ( 1st ` <. x , y >. ) = x
14	12, 13	syl6req 2089	. . . . . . . 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 5182	. . . . . . . . 9 |- ( A = <. <. x , y >. , z >. -> ( 2nd ` ( 1st ` A ) ) = ( 2nd ` <. x , y >. ) )
18	7, 8	op2nd 5774	. . . . . . . . 9 |- ( 2nd ` <. x , y >. ) = y
19	17, 18	syl6req 2089	. . . . . . . 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 5776	. . . . . . . . 9 |- ( A = <. <. x , y >. , z >. -> ( 2nd ` A ) = z )
23	22	eqcomd 2045	. . . . . . . 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 203	. . . . . 6 |- ( A = <. <. x , y >. , z >. -> ( ph <-> th ) )
27	26	biimpa 280	. . . . 5 |- ( ( A = <. <. x , y >. , z >. /\ ph ) -> th )
28	27	exlimiv 1489	. . . 4 |- ( E. z ( A = <. <. x , y >. , z >. /\ ph ) -> th )
29	28	exlimiv 1489	. . 3 |- ( E. y E. z ( A = <. <. x , y >. , z >. /\ ph ) -> th )
30	29	exlimiv 1489	. 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 97   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393   <.cop 3378   ` cfv 4902   {coprab 5513   1stc1st 5765   2ndc2nd 5766

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fv 4910  df-oprab 5516  df-1st 5767  df-2nd 5768

This theorem is referenced by: (None)