Theorem opabex3d 6264

Description: Existence of an ordered pair abstraction, deduction version. (Contributed by Alexander van der Vekens, 19-Oct-2017.)

Hypotheses

Ref	Expression
opabex3d.1	|- ( ph -> A e. _V )
opabex3d.2	|- ( ( ph /\ x e. A ) -> { y | ps } e. _V )

Assertion

Ref	Expression
opabex3d	|- ( ph -> { <. x , y >. | ( x e. A /\ ps ) } e. _V )

Distinct variable groups:   x, A, y   ph, x

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

Proof of Theorem opabex3d

Dummy variables v w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		19.42v 1953	. . . . . 6 |- ( E. y ( x e. A /\ ( z = <. x , y >. /\ ps ) ) <-> ( x e. A /\ E. y ( z = <. x , y >. /\ ps ) ) )
2		an12 561	. . . . . . 7 |- ( ( z = <. x , y >. /\ ( x e. A /\ ps ) ) <-> ( x e. A /\ ( z = <. x , y >. /\ ps ) ) )
3	2	exbii 1651	. . . . . 6 |- ( E. y ( z = <. x , y >. /\ ( x e. A /\ ps ) ) <-> E. y ( x e. A /\ ( z = <. x , y >. /\ ps ) ) )
4		elxp 4735	. . . . . . . 8 |- ( z e. ( { x } X. { y | ps } ) <-> E. v E. w ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ps } ) ) )
5		excom 1710	. . . . . . . . 9 |- ( E. v E. w ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ps } ) ) <-> E. w E. v ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ps } ) ) )
6		an12 561	. . . . . . . . . . . . 13 |- ( ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ps } ) ) <-> ( v e. { x } /\ ( z = <. v , w >. /\ w e. { y | ps } ) ) )
7		velsn 3683	. . . . . . . . . . . . . 14 |- ( v e. { x } <-> v = x )
8	7	anbi1i 458	. . . . . . . . . . . . 13 |- ( ( v e. { x } /\ ( z = <. v , w >. /\ w e. { y | ps } ) ) <-> ( v = x /\ ( z = <. v , w >. /\ w e. { y | ps } ) ) )
9	6, 8	bitri 184	. . . . . . . . . . . 12 |- ( ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ps } ) ) <-> ( v = x /\ ( z = <. v , w >. /\ w e. { y | ps } ) ) )
10	9	exbii 1651	. . . . . . . . . . 11 |- ( E. v ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ps } ) ) <-> E. v ( v = x /\ ( z = <. v , w >. /\ w e. { y | ps } ) ) )
11		vex 2802	. . . . . . . . . . . 12 |- x e. _V
12		opeq1 3856	. . . . . . . . . . . . . 14 |- ( v = x -> <. v , w >. = <. x , w >. )
13	12	eqeq2d 2241	. . . . . . . . . . . . 13 |- ( v = x -> ( z = <. v , w >. <-> z = <. x , w >. ) )
14	13	anbi1d 465	. . . . . . . . . . . 12 |- ( v = x -> ( ( z = <. v , w >. /\ w e. { y | ps } ) <-> ( z = <. x , w >. /\ w e. { y | ps } ) ) )
15	11, 14	ceqsexv 2839	. . . . . . . . . . 11 |- ( E. v ( v = x /\ ( z = <. v , w >. /\ w e. { y | ps } ) ) <-> ( z = <. x , w >. /\ w e. { y | ps } ) )
16	10, 15	bitri 184	. . . . . . . . . 10 |- ( E. v ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ps } ) ) <-> ( z = <. x , w >. /\ w e. { y | ps } ) )
17	16	exbii 1651	. . . . . . . . 9 |- ( E. w E. v ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ps } ) ) <-> E. w ( z = <. x , w >. /\ w e. { y | ps } ) )
18	5, 17	bitri 184	. . . . . . . 8 |- ( E. v E. w ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ps } ) ) <-> E. w ( z = <. x , w >. /\ w e. { y | ps } ) )
19		nfv 1574	. . . . . . . . . 10 |- F/ y z = <. x , w >.
20		nfsab1 2219	. . . . . . . . . 10 |- F/ y w e. { y | ps }
21	19, 20	nfan 1611	. . . . . . . . 9 |- F/ y ( z = <. x , w >. /\ w e. { y | ps } )
22		nfv 1574	. . . . . . . . 9 |- F/ w ( z = <. x , y >. /\ ps )
23		opeq2 3857	. . . . . . . . . . 11 |- ( w = y -> <. x , w >. = <. x , y >. )
24	23	eqeq2d 2241	. . . . . . . . . 10 |- ( w = y -> ( z = <. x , w >. <-> z = <. x , y >. ) )
25		df-clab 2216	. . . . . . . . . . 11 |- ( w e. { y | ps } <-> [ w / y ] ps )
26		sbequ12 1817	. . . . . . . . . . . 12 |- ( y = w -> ( ps <-> [ w / y ] ps ) )
27	26	equcoms 1754	. . . . . . . . . . 11 |- ( w = y -> ( ps <-> [ w / y ] ps ) )
28	25, 27	bitr4id 199	. . . . . . . . . 10 |- ( w = y -> ( w e. { y | ps } <-> ps ) )
29	24, 28	anbi12d 473	. . . . . . . . 9 |- ( w = y -> ( ( z = <. x , w >. /\ w e. { y | ps } ) <-> ( z = <. x , y >. /\ ps ) ) )
30	21, 22, 29	cbvex 1802	. . . . . . . 8 |- ( E. w ( z = <. x , w >. /\ w e. { y | ps } ) <-> E. y ( z = <. x , y >. /\ ps ) )
31	4, 18, 30	3bitri 206	. . . . . . 7 |- ( z e. ( { x } X. { y | ps } ) <-> E. y ( z = <. x , y >. /\ ps ) )
32	31	anbi2i 457	. . . . . 6 |- ( ( x e. A /\ z e. ( { x } X. { y | ps } ) ) <-> ( x e. A /\ E. y ( z = <. x , y >. /\ ps ) ) )
33	1, 3, 32	3bitr4ri 213	. . . . 5 |- ( ( x e. A /\ z e. ( { x } X. { y | ps } ) ) <-> E. y ( z = <. x , y >. /\ ( x e. A /\ ps ) ) )
34	33	exbii 1651	. . . 4 |- ( E. x ( x e. A /\ z e. ( { x } X. { y | ps } ) ) <-> E. x E. y ( z = <. x , y >. /\ ( x e. A /\ ps ) ) )
35		eliun 3968	. . . . 5 |- ( z e. U_ x e. A ( { x } X. { y | ps } ) <-> E. x e. A z e. ( { x } X. { y | ps } ) )
36		df-rex 2514	. . . . 5 |- ( E. x e. A z e. ( { x } X. { y | ps } ) <-> E. x ( x e. A /\ z e. ( { x } X. { y | ps } ) ) )
37	35, 36	bitri 184	. . . 4 |- ( z e. U_ x e. A ( { x } X. { y | ps } ) <-> E. x ( x e. A /\ z e. ( { x } X. { y | ps } ) ) )
38		elopab 4345	. . . 4 |- ( z e. { <. x , y >. | ( x e. A /\ ps ) } <-> E. x E. y ( z = <. x , y >. /\ ( x e. A /\ ps ) ) )
39	34, 37, 38	3bitr4i 212	. . 3 |- ( z e. U_ x e. A ( { x } X. { y | ps } ) <-> z e. { <. x , y >. | ( x e. A /\ ps ) } )
40	39	eqriv 2226	. 2 |- U_ x e. A ( { x } X. { y | ps } ) = { <. x , y >. | ( x e. A /\ ps ) }
41		opabex3d.1	. . 3 |- ( ph -> A e. _V )
42		snexg 4267	. . . . . 6 |- ( x e. _V -> { x } e. _V )
43	11, 42	ax-mp 5	. . . . 5 |- { x } e. _V
44		opabex3d.2	. . . . 5 |- ( ( ph /\ x e. A ) -> { y | ps } e. _V )
45		xpexg 4832	. . . . 5 |- ( ( { x } e. _V /\ { y | ps } e. _V ) -> ( { x } X. { y | ps } ) e. _V )
46	43, 44, 45	sylancr 414	. . . 4 |- ( ( ph /\ x e. A ) -> ( { x } X. { y | ps } ) e. _V )
47	46	ralrimiva 2603	. . 3 |- ( ph -> A. x e. A ( { x } X. { y | ps } ) e. _V )
48		iunexg 6262	. . 3 |- ( ( A e. _V /\ A. x e. A ( { x } X. { y | ps } ) e. _V ) -> U_ x e. A ( { x } X. { y | ps } ) e. _V )
49	41, 47, 48	syl2anc 411	. 2 |- ( ph -> U_ x e. A ( { x } X. { y | ps } ) e. _V )
50	40, 49	eqeltrrid 2317	1 |- ( ph -> { <. x , y >. | ( x e. A /\ ps ) } e. _V )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   E.wex 1538   [wsb 1808   e. wcel 2200   {cab 2215   A.wral 2508   E.wrex 2509   _Vcvv 2799   {csn 3666   <.cop 3669   U_ciun 3964   {copab 4143   X. cxp 4716

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325

This theorem is referenced by:  acfun  7385  ccfunen  7446  ovshftex  11325  wksfval  16028