Theorem dfoprab4f 5761

Description: Operation class abstraction expressed without existential quantifiers. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypotheses

Ref	Expression
dfoprab4f.x	|- F/xph
dfoprab4f.y	|- F/yph
dfoprab4f.1	|- (w = <.x , y >. -> (ph <-> ps))

Assertion

Ref	Expression
dfoprab4f	|- { <.w , z >. | (w e. (A X. B) /\ ph) } = { <. <.x , y >. , z >. | ((x e. A /\ y e. B) /\ ps) }

Distinct variable groups:   x,w,y,z   w,A,x,y   w,B,x,y   ps,w

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

Proof of Theorem dfoprab4f

Dummy variables u t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1418	. . . . 5 |- F/x w = <. t , u >.
2		dfoprab4f.x	. . . . . 6 |- F/xph
3		nfs1v 1812	. . . . . 6 |- F/x[ t / x][u / y]ps
4	2, 3	nfbi 1478	. . . . 5 |- F/x(ph <-> [ t / x][u / y]ps)
5	1, 4	nfim 1461	. . . 4 |- F/x(w = <. t , u >. -> (ph <-> [ t / x][u / y]ps))
6		opeq1 3540	. . . . . 6 |- (x = t -> <.x , u >. = <. t , u >.)
7	6	eqeq2d 2048	. . . . 5 |- (x = t -> (w = <.x , u >. <-> w = <. t , u >.))
8		sbequ12 1651	. . . . . 6 |- (x = t -> ([u / y]ps <-> [ t / x][u / y]ps))
9	8	bibi2d 221	. . . . 5 |- (x = t -> ((ph <-> [u / y]ps) <-> (ph <-> [ t / x][u / y]ps)))
10	7, 9	imbi12d 223	. . . 4 |- (x = t -> ((w = <.x , u >. -> (ph <-> [u / y]ps)) <-> (w = <. t , u >. -> (ph <-> [ t / x][u / y]ps))))
11		nfv 1418	. . . . . 6 |- F/y w = <.x , u >.
12		dfoprab4f.y	. . . . . . 7 |- F/yph
13		nfs1v 1812	. . . . . . 7 |- F/y[u / y]ps
14	12, 13	nfbi 1478	. . . . . 6 |- F/y(ph <-> [u / y]ps)
15	11, 14	nfim 1461	. . . . 5 |- F/y(w = <.x , u >. -> (ph <-> [u / y]ps))
16		opeq2 3541	. . . . . . 7 |- (y = u -> <.x , y >. = <.x , u >.)
17	16	eqeq2d 2048	. . . . . 6 |- (y = u -> (w = <.x , y >. <-> w = <.x , u >.))
18		sbequ12 1651	. . . . . . 7 |- (y = u -> (ps <-> [u / y]ps))
19	18	bibi2d 221	. . . . . 6 |- (y = u -> ((ph <-> ps) <-> (ph <-> [u / y]ps)))
20	17, 19	imbi12d 223	. . . . 5 |- (y = u -> ((w = <.x , y >. -> (ph <-> ps)) <-> (w = <.x , u >. -> (ph <-> [u / y]ps))))
21		dfoprab4f.1	. . . . 5 |- (w = <.x , y >. -> (ph <-> ps))
22	15, 20, 21	chvar 1637	. . . 4 |- (w = <.x , u >. -> (ph <-> [u / y]ps))
23	5, 10, 22	chvar 1637	. . 3 |- (w = <. t , u >. -> (ph <-> [ t / x][u / y]ps))
24	23	dfoprab4 5760	. 2 |- { <.w , z >. | (w e. (A X. B) /\ ph) } = { <. <. t , u >. , z >. | (( t e. A /\ u e. B) /\ [ t / x][u / y]ps) }
25		nfv 1418	. . 3 |- F/ t((x e. A /\ y e. B) /\ ps)
26		nfv 1418	. . 3 |- F/u((x e. A /\ y e. B) /\ ps)
27		nfv 1418	. . . 4 |- F/x( t e. A /\ u e. B)
28	27, 3	nfan 1454	. . 3 |- F/x(( t e. A /\ u e. B) /\ [ t / x][u / y]ps)
29		nfv 1418	. . . 4 |- F/y( t e. A /\ u e. B)
30	13	nfsb 1819	. . . 4 |- F/y[ t / x][u / y]ps
31	29, 30	nfan 1454	. . 3 |- F/y(( t e. A /\ u e. B) /\ [ t / x][u / y]ps)
32		eleq1 2097	. . . . 5 |- (x = t -> (x e. A <-> t e. A))
33		eleq1 2097	. . . . 5 |- (y = u -> (y e. B <-> u e. B))
34	32, 33	bi2anan9 538	. . . 4 |- ((x = t /\ y = u) -> ((x e. A /\ y e. B) <-> ( t e. A /\ u e. B)))
35	18, 8	sylan9bbr 436	. . . 4 |- ((x = t /\ y = u) -> (ps <-> [ t / x][u / y]ps))
36	34, 35	anbi12d 442	. . 3 |- ((x = t /\ y = u) -> (((x e. A /\ y e. B) /\ ps) <-> (( t e. A /\ u e. B) /\ [ t / x][u / y]ps)))
37	25, 26, 28, 31, 36	cbvoprab12 5520	. 2 |- { <. <.x , y >. , z >. | ((x e. A /\ y e. B) /\ ps) } = { <. <. t , u >. , z >. | (( t e. A /\ u e. B) /\ [ t / x][u / y]ps) }
38	24, 37	eqtr4i 2060	1 |- { <.w , z >. | (w e. (A X. B) /\ ph) } = { <. <.x , y >. , z >. | ((x e. A /\ y e. B) /\ ps) }

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1242   F/wnf 1346   e. wcel 1390  [wsb 1642   <.cop 3370   {copab 3808   X. cxp 4286   {coprab 5456

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: (None)