Theorem dfoprab3 5759

Description: Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 16-Dec-2008.)

Hypothesis

Ref	Expression
dfoprab3.1	|- (w = <.x , y >. -> (ph <-> ps))

Assertion

Ref	Expression
dfoprab3	|- { <.w , z >. | (w e. ( _V X. _V) /\ ph) } = { <. <.x , y >. , z >. | ps }

Distinct variable groups:   x,y,ph   ps,w   x,z,w,y

Allowed substitution hints:   ph(z,w)   ps(x,y,z)

Proof of Theorem dfoprab3

Step	Hyp	Ref	Expression
1		dfoprab3s 5758	. 2 |- { <. <.x , y >. , z >. | ps } = { <.w , z >. | (w e. ( _V X. _V) /\ [.( 1st ` w) / x ]. [.( 2nd ` w) / y ].ps) }
2		vex 2554	. . . . . 6 |- w e. _V
3		1stexg 5736	. . . . . 6 |- (w e. _V -> ( 1st ` w) e. _V)
4	2, 3	ax-mp 7	. . . . 5 |- ( 1st ` w) e. _V
5		2ndexg 5737	. . . . . 6 |- (w e. _V -> ( 2nd ` w) e. _V)
6	2, 5	ax-mp 7	. . . . 5 |- ( 2nd ` w) e. _V
7		eqcom 2039	. . . . . . . . . 10 |- (x = ( 1st ` w) <-> ( 1st ` w) = x)
8		eqcom 2039	. . . . . . . . . 10 |- (y = ( 2nd ` w) <-> ( 2nd ` w) = y)
9	7, 8	anbi12i 433	. . . . . . . . 9 |- ((x = ( 1st ` w) /\ y = ( 2nd ` w)) <-> (( 1st ` w) = x /\ ( 2nd ` w) = y))
10		eqopi 5740	. . . . . . . . 9 |- ((w e. ( _V X. _V) /\ (( 1st ` w) = x /\ ( 2nd ` w) = y)) -> w = <.x , y >.)
11	9, 10	sylan2b 271	. . . . . . . 8 |- ((w e. ( _V X. _V) /\ (x = ( 1st ` w) /\ y = ( 2nd ` w))) -> w = <.x , y >.)
12		dfoprab3.1	. . . . . . . 8 |- (w = <.x , y >. -> (ph <-> ps))
13	11, 12	syl 14	. . . . . . 7 |- ((w e. ( _V X. _V) /\ (x = ( 1st ` w) /\ y = ( 2nd ` w))) -> (ph <-> ps))
14	13	bicomd 129	. . . . . 6 |- ((w e. ( _V X. _V) /\ (x = ( 1st ` w) /\ y = ( 2nd ` w))) -> (ps <-> ph))
15	14	ex 108	. . . . 5 |- (w e. ( _V X. _V) -> ((x = ( 1st ` w) /\ y = ( 2nd ` w)) -> (ps <-> ph)))
16	4, 6, 15	sbc2iedv 2824	. . . 4 |- (w e. ( _V X. _V) -> ( [.( 1st ` w) / x ]. [.( 2nd ` w) / y ].ps <-> ph))
17	16	pm5.32i 427	. . 3 |- ((w e. ( _V X. _V) /\ [.( 1st ` w) / x ]. [.( 2nd ` w) / y ].ps) <-> (w e. ( _V X. _V) /\ ph))
18	17	opabbii 3815	. 2 |- { <.w , z >. | (w e. ( _V X. _V) /\ [.( 1st ` w) / x ]. [.( 2nd ` w) / y ].ps) } = { <.w , z >. | (w e. ( _V X. _V) /\ ph) }
19	1, 18	eqtr2i 2058	1 |- { <.w , z >. | (w e. ( _V X. _V) /\ ph) } = { <. <.x , y >. , z >. | ps }

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1242   e. wcel 1390   _Vcvv 2551   [.wsbc 2758   <.cop 3370   {copab 3808   X. cxp 4286   ` 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-fn 4848  df-f 4849  df-fo 4851  df-fv 4853  df-oprab 5459  df-1st 5709  df-2nd 5710

This theorem is referenced by:  dfoprab4  5760  df1st2  5782  df2nd2  5783