Theorem eloprabi 5764

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 2043	. . . . . 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 1746	. . . 4 |- (w = A -> (E.xE.yE.z(w = <. <.x , y >. , z >. /\ ph) <-> E.xE.yE.z(A = <. <.x , y >. , z >. /\ ph)))
4		df-oprab 5459	. . . 4 |- { <. <.x , y >. , z >. | ph } = {w | E.xE.yE.z(w = <. <.x , y >. , z >. /\ ph) }
5	3, 4	elab2g 2683	. . 3 |- (A e. { <. <.x , y >. , z >. | ph } -> (A e. { <. <.x , y >. , z >. | ph } <-> E.xE.yE.z(A = <. <.x , y >. , z >. /\ ph)))
6	5	ibi 165	. 2 |- (A e. { <. <.x , y >. , z >. | ph } -> E.xE.yE.z(A = <. <.x , y >. , z >. /\ ph))
7		vex 2554	. . . . . . . . . . . 12 |- x e. _V
8		vex 2554	. . . . . . . . . . . 12 |- y e. _V
9	7, 8	opex 3957	. . . . . . . . . . 11 |- <.x , y >. e. _V
10		vex 2554	. . . . . . . . . . 11 |- z e. _V
11	9, 10	op1std 5717	. . . . . . . . . 10 |- (A = <. <.x , y >. , z >. -> ( 1st ` A) = <.x , y >.)
12	11	fveq2d 5125	. . . . . . . . 9 |- (A = <. <.x , y >. , z >. -> ( 1st ` ( 1st ` A)) = ( 1st ` <.x , y >.))
13	7, 8	op1st 5715	. . . . . . . . 9 |- ( 1st ` <.x , y >.) = x
14	12, 13	syl6req 2086	. . . . . . . 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 5125	. . . . . . . . 9 |- (A = <. <.x , y >. , z >. -> ( 2nd ` ( 1st ` A)) = ( 2nd ` <.x , y >.))
18	7, 8	op2nd 5716	. . . . . . . . 9 |- ( 2nd ` <.x , y >.) = y
19	17, 18	syl6req 2086	. . . . . . . 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 5718	. . . . . . . . 9 |- (A = <. <.x , y >. , z >. -> ( 2nd ` A) = z)
23	22	eqcomd 2042	. . . . . . . 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 1486	. . . 4 |- (E.z(A = <. <.x , y >. , z >. /\ ph) -> th)
29	28	exlimiv 1486	. . 3 |- (E.yE.z(A = <. <.x , y >. , z >. /\ ph) -> th)
30	29	exlimiv 1486	. 2 |- (E.xE.yE.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 1242  E.wex 1378   e. wcel 1390   <.cop 3370   ` cfv 4845   {coprab 5456   1stc1st 5707   2ndc2nd 5708

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fv 4853  df-oprab 5459  df-1st 5709  df-2nd 5710

This theorem is referenced by: (None)