Theorem eloprabi 7232

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 2626	. . . . . 6 |- ( w = A -> ( w = <. <. x , y >. , z >. <-> A = <. <. x , y >. , z >. ) )
2	1	anbi1d 741	. . . . 5 |- ( w = A -> ( ( w = <. <. x , y >. , z >. /\ ph ) <-> ( A = <. <. x , y >. , z >. /\ ph ) ) )
3	2	3exbidv 1853	. . . 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 6654	. . . 4 |- { <. <. x , y >. , z >. | ph } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) }
5	3, 4	elab2g 3353	. . 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 256	. 2 |- ( A e. { <. <. x , y >. , z >. | ph } -> E. x E. y E. z ( A = <. <. x , y >. , z >. /\ ph ) )
7		opex 4932	. . . . . . . . . . 11 |- <. x , y >. e. _V
8		vex 3203	. . . . . . . . . . 11 |- z e. _V
9	7, 8	op1std 7178	. . . . . . . . . 10 |- ( A = <. <. x , y >. , z >. -> ( 1st ` A ) = <. x , y >. )
10	9	fveq2d 6195	. . . . . . . . 9 |- ( A = <. <. x , y >. , z >. -> ( 1st ` ( 1st ` A ) ) = ( 1st ` <. x , y >. ) )
11		vex 3203	. . . . . . . . . 10 |- x e. _V
12		vex 3203	. . . . . . . . . 10 |- y e. _V
13	11, 12	op1st 7176	. . . . . . . . 9 |- ( 1st ` <. x , y >. ) = x
14	10, 13	syl6req 2673	. . . . . . . 8 |- ( A = <. <. x , y >. , z >. -> x = ( 1st ` ( 1st ` A ) ) )
15		eloprabi.1	. . . . . . . 8 |- ( x = ( 1st ` ( 1st ` A ) ) -> ( ph <-> ps ) )
16	14, 15	syl 17	. . . . . . 7 |- ( A = <. <. x , y >. , z >. -> ( ph <-> ps ) )
17	9	fveq2d 6195	. . . . . . . . 9 |- ( A = <. <. x , y >. , z >. -> ( 2nd ` ( 1st ` A ) ) = ( 2nd ` <. x , y >. ) )
18	11, 12	op2nd 7177	. . . . . . . . 9 |- ( 2nd ` <. x , y >. ) = y
19	17, 18	syl6req 2673	. . . . . . . 8 |- ( A = <. <. x , y >. , z >. -> y = ( 2nd ` ( 1st ` A ) ) )
20		eloprabi.2	. . . . . . . 8 |- ( y = ( 2nd ` ( 1st ` A ) ) -> ( ps <-> ch ) )
21	19, 20	syl 17	. . . . . . 7 |- ( A = <. <. x , y >. , z >. -> ( ps <-> ch ) )
22	7, 8	op2ndd 7179	. . . . . . . . 9 |- ( A = <. <. x , y >. , z >. -> ( 2nd ` A ) = z )
23	22	eqcomd 2628	. . . . . . . 8 |- ( A = <. <. x , y >. , z >. -> z = ( 2nd ` A ) )
24		eloprabi.3	. . . . . . . 8 |- ( z = ( 2nd ` A ) -> ( ch <-> th ) )
25	23, 24	syl 17	. . . . . . 7 |- ( A = <. <. x , y >. , z >. -> ( ch <-> th ) )
26	16, 21, 25	3bitrd 294	. . . . . 6 |- ( A = <. <. x , y >. , z >. -> ( ph <-> th ) )
27	26	biimpa 501	. . . . 5 |- ( ( A = <. <. x , y >. , z >. /\ ph ) -> th )
28	27	exlimiv 1858	. . . 4 |- ( E. z ( A = <. <. x , y >. , z >. /\ ph ) -> th )
29	28	exlimiv 1858	. . 3 |- ( E. y E. z ( A = <. <. x , y >. , z >. /\ ph ) -> th )
30	29	exlimiv 1858	. 2 |- ( E. x E. y E. z ( A = <. <. x , y >. , z >. /\ ph ) -> th )
31	6, 30	syl 17	1 |- ( A e. { <. <. x , y >. , z >. | ph } -> th )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   <.cop 4183   ` cfv 5888   {coprab 6651   1stc1st 7166   2ndc2nd 7167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-oprab 6654  df-1st 7168  df-2nd 7169

This theorem is referenced by: (None)