Theorem eloprabi 6370

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 2238	. . . . . 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 1917	. . . 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 6032	. . . 4 |- { <. <. x , y >. , z >. | ph } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) }
5	3, 4	elab2g 2954	. . 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 2806	. . . . . . . . . . . 12 |- x e. _V
8		vex 2806	. . . . . . . . . . . 12 |- y e. _V
9	7, 8	opex 4327	. . . . . . . . . . 11 |- <. x , y >. e. _V
10		vex 2806	. . . . . . . . . . 11 |- z e. _V
11	9, 10	op1std 6320	. . . . . . . . . 10 |- ( A = <. <. x , y >. , z >. -> ( 1st ` A ) = <. x , y >. )
12	11	fveq2d 5652	. . . . . . . . 9 |- ( A = <. <. x , y >. , z >. -> ( 1st ` ( 1st ` A ) ) = ( 1st ` <. x , y >. ) )
13	7, 8	op1st 6318	. . . . . . . . 9 |- ( 1st ` <. x , y >. ) = x
14	12, 13	eqtr2di 2281	. . . . . . . 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 5652	. . . . . . . . 9 |- ( A = <. <. x , y >. , z >. -> ( 2nd ` ( 1st ` A ) ) = ( 2nd ` <. x , y >. ) )
18	7, 8	op2nd 6319	. . . . . . . . 9 |- ( 2nd ` <. x , y >. ) = y
19	17, 18	eqtr2di 2281	. . . . . . . 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 6321	. . . . . . . . 9 |- ( A = <. <. x , y >. , z >. -> ( 2nd ` A ) = z )
23	22	eqcomd 2237	. . . . . . . 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 1647	. . . 4 |- ( E. z ( A = <. <. x , y >. , z >. /\ ph ) -> th )
29	28	exlimiv 1647	. . 3 |- ( E. y E. z ( A = <. <. x , y >. , z >. /\ ph ) -> th )
30	29	exlimiv 1647	. 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 1398   E.wex 1541   e. wcel 2202   <.cop 3676   ` cfv 5333   {coprab 6029   1stc1st 6310   2ndc2nd 6311

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fv 5341  df-oprab 6032  df-1st 6312  df-2nd 6313

This theorem is referenced by: (None)