Theorem opabex3d 6558

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 1924	. . . . . 6 |- ( E. y ( x e. A /\ ( z = <. x , y >. /\ ps ) ) <-> ( x e. A /\ E. y ( z = <. x , y >. /\ ps ) ) )
2		an12 795	. . . . . . 7 |- ( ( z = <. x , y >. /\ ( x e. A /\ ps ) ) <-> ( x e. A /\ ( z = <. x , y >. /\ ps ) ) )
3	2	exbii 1634	. . . . . 6 |- ( E. y ( z = <. x , y >. /\ ( x e. A /\ ps ) ) <-> E. y ( x e. A /\ ( z = <. x , y >. /\ ps ) ) )
4		elxp 4860	. . . . . . . 8 |- ( z e. ( { x } X. { y | ps } ) <-> E. v E. w ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ps } ) ) )
5		excom 1787	. . . . . . . . 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 795	. . . . . . . . . . . . 13 |- ( ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ps } ) ) <-> ( v e. { x } /\ ( z = <. v , w >. /\ w e. { y | ps } ) ) )
7		elsn 3894	. . . . . . . . . . . . . 14 |- ( v e. { x } <-> v = x )
8	7	anbi1i 695	. . . . . . . . . . . . 13 |- ( ( v e. { x } /\ ( z = <. v , w >. /\ w e. { y | ps } ) ) <-> ( v = x /\ ( z = <. v , w >. /\ w e. { y | ps } ) ) )
9	6, 8	bitri 249	. . . . . . . . . . . 12 |- ( ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ps } ) ) <-> ( v = x /\ ( z = <. v , w >. /\ w e. { y | ps } ) ) )
10	9	exbii 1634	. . . . . . . . . . 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 2978	. . . . . . . . . . . 12 |- x e. _V
12		opeq1 4062	. . . . . . . . . . . . . 14 |- ( v = x -> <. v , w >. = <. x , w >. )
13	12	eqeq2d 2454	. . . . . . . . . . . . 13 |- ( v = x -> ( z = <. v , w >. <-> z = <. x , w >. ) )
14	13	anbi1d 704	. . . . . . . . . . . 12 |- ( v = x -> ( ( z = <. v , w >. /\ w e. { y | ps } ) <-> ( z = <. x , w >. /\ w e. { y | ps } ) ) )
15	11, 14	ceqsexv 3012	. . . . . . . . . . 11 |- ( E. v ( v = x /\ ( z = <. v , w >. /\ w e. { y | ps } ) ) <-> ( z = <. x , w >. /\ w e. { y | ps } ) )
16	10, 15	bitri 249	. . . . . . . . . 10 |- ( E. v ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ps } ) ) <-> ( z = <. x , w >. /\ w e. { y | ps } ) )
17	16	exbii 1634	. . . . . . . . 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 249	. . . . . . . 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 1673	. . . . . . . . . 10 |- F/ y z = <. x , w >.
20		nfsab1 2433	. . . . . . . . . 10 |- F/ y w e. { y | ps }
21	19, 20	nfan 1861	. . . . . . . . 9 |- F/ y ( z = <. x , w >. /\ w e. { y | ps } )
22		nfv 1673	. . . . . . . . 9 |- F/ w ( z = <. x , y >. /\ ps )
23		opeq2 4063	. . . . . . . . . . 11 |- ( w = y -> <. x , w >. = <. x , y >. )
24	23	eqeq2d 2454	. . . . . . . . . 10 |- ( w = y -> ( z = <. x , w >. <-> z = <. x , y >. ) )
25		sbequ12 1936	. . . . . . . . . . . 12 |- ( y = w -> ( ps <-> [ w / y ] ps ) )
26	25	equcoms 1733	. . . . . . . . . . 11 |- ( w = y -> ( ps <-> [ w / y ] ps ) )
27		df-clab 2430	. . . . . . . . . . 11 |- ( w e. { y | ps } <-> [ w / y ] ps )
28	26, 27	syl6rbbr 264	. . . . . . . . . 10 |- ( w = y -> ( w e. { y | ps } <-> ps ) )
29	24, 28	anbi12d 710	. . . . . . . . 9 |- ( w = y -> ( ( z = <. x , w >. /\ w e. { y | ps } ) <-> ( z = <. x , y >. /\ ps ) ) )
30	21, 22, 29	cbvex 1970	. . . . . . . 8 |- ( E. w ( z = <. x , w >. /\ w e. { y | ps } ) <-> E. y ( z = <. x , y >. /\ ps ) )
31	4, 18, 30	3bitri 271	. . . . . . 7 |- ( z e. ( { x } X. { y | ps } ) <-> E. y ( z = <. x , y >. /\ ps ) )
32	31	anbi2i 694	. . . . . 6 |- ( ( x e. A /\ z e. ( { x } X. { y | ps } ) ) <-> ( x e. A /\ E. y ( z = <. x , y >. /\ ps ) ) )
33	1, 3, 32	3bitr4ri 278	. . . . 5 |- ( ( x e. A /\ z e. ( { x } X. { y | ps } ) ) <-> E. y ( z = <. x , y >. /\ ( x e. A /\ ps ) ) )
34	33	exbii 1634	. . . 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 4178	. . . . 5 |- ( z e. U_ x e. A ( { x } X. { y | ps } ) <-> E. x e. A z e. ( { x } X. { y | ps } ) )
36		df-rex 2724	. . . . 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 249	. . . 4 |- ( z e. U_ x e. A ( { x } X. { y | ps } ) <-> E. x ( x e. A /\ z e. ( { x } X. { y | ps } ) ) )
38		elopab 4600	. . . 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 277	. . 3 |- ( z e. U_ x e. A ( { x } X. { y | ps } ) <-> z e. { <. x , y >. | ( x e. A /\ ps ) } )
40	39	eqriv 2440	. 2 |- U_ x e. A ( { x } X. { y | ps } ) = { <. x , y >. | ( x e. A /\ ps ) }
41		opabex3d.1	. . 3 |- ( ph -> A e. _V )
42		snex 4536	. . . . 5 |- { x } e. _V
43		opabex3d.2	. . . . 5 |- ( ( ph /\ x e. A ) -> { y | ps } e. _V )
44		xpexg 6510	. . . . 5 |- ( ( { x } e. _V /\ { y | ps } e. _V ) -> ( { x } X. { y | ps } ) e. _V )
45	42, 43, 44	sylancr 663	. . . 4 |- ( ( ph /\ x e. A ) -> ( { x } X. { y | ps } ) e. _V )
46	45	ralrimiva 2802	. . 3 |- ( ph -> A. x e. A ( { x } X. { y | ps } ) e. _V )
47		iunexg 6556	. . 3 |- ( ( A e. _V /\ A. x e. A ( { x } X. { y | ps } ) e. _V ) -> U_ x e. A ( { x } X. { y | ps } ) e. _V )
48	41, 46, 47	syl2anc 661	. 2 |- ( ph -> U_ x e. A ( { x } X. { y | ps } ) e. _V )
49	40, 48	syl5eqelr 2528	1 |- ( ph -> { <. x , y >. | ( x e. A /\ ps ) } e. _V )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1369   E.wex 1586   [wsb 1700   e. wcel 1756   {cab 2429   A.wral 2718   E.wrex 2719   _Vcvv 2975   {csn 3880   <.cop 3886   U_ciun 4174   {copab 4352   X. cxp 4841

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-ral 2723  df-rex 2724  df-reu 2725  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-op 3887  df-uni 4095  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-id 4639  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429

This theorem is referenced by:  wlks  23428  wlkres  23431  trls  23438  crcts  23511  cycls  23512  fpwrelmap  26036