Theorem opabex3d 6724

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 1827	. . . . . 6 |- ( E. y ( x e. A /\ ( z = <. x , y >. /\ ps ) ) <-> ( x e. A /\ E. y ( z = <. x , y >. /\ ps ) ) )
2		an12 804	. . . . . . 7 |- ( ( z = <. x , y >. /\ ( x e. A /\ ps ) ) <-> ( x e. A /\ ( z = <. x , y >. /\ ps ) ) )
3	2	exbii 1712	. . . . . 6 |- ( E. y ( z = <. x , y >. /\ ( x e. A /\ ps ) ) <-> E. y ( x e. A /\ ( z = <. x , y >. /\ ps ) ) )
4		elxp 4808	. . . . . . . 8 |- ( z e. ( { x } X. { y | ps } ) <-> E. v E. w ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ps } ) ) )
5		excom 1903	. . . . . . . . 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 804	. . . . . . . . . . . . 13 |- ( ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ps } ) ) <-> ( v e. { x } /\ ( z = <. v , w >. /\ w e. { y | ps } ) ) )
7		elsn 3950	. . . . . . . . . . . . . 14 |- ( v e. { x } <-> v = x )
8	7	anbi1i 699	. . . . . . . . . . . . 13 |- ( ( v e. { x } /\ ( z = <. v , w >. /\ w e. { y | ps } ) ) <-> ( v = x /\ ( z = <. v , w >. /\ w e. { y | ps } ) ) )
9	6, 8	bitri 252	. . . . . . . . . . . 12 |- ( ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ps } ) ) <-> ( v = x /\ ( z = <. v , w >. /\ w e. { y | ps } ) ) )
10	9	exbii 1712	. . . . . . . . . . 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 3020	. . . . . . . . . . . 12 |- x e. _V
12		opeq1 4125	. . . . . . . . . . . . . 14 |- ( v = x -> <. v , w >. = <. x , w >. )
13	12	eqeq2d 2433	. . . . . . . . . . . . 13 |- ( v = x -> ( z = <. v , w >. <-> z = <. x , w >. ) )
14	13	anbi1d 709	. . . . . . . . . . . 12 |- ( v = x -> ( ( z = <. v , w >. /\ w e. { y | ps } ) <-> ( z = <. x , w >. /\ w e. { y | ps } ) ) )
15	11, 14	ceqsexv 3055	. . . . . . . . . . 11 |- ( E. v ( v = x /\ ( z = <. v , w >. /\ w e. { y | ps } ) ) <-> ( z = <. x , w >. /\ w e. { y | ps } ) )
16	10, 15	bitri 252	. . . . . . . . . 10 |- ( E. v ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ps } ) ) <-> ( z = <. x , w >. /\ w e. { y | ps } ) )
17	16	exbii 1712	. . . . . . . . 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 252	. . . . . . . 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 1755	. . . . . . . . . 10 |- F/ y z = <. x , w >.
20		nfsab1 2413	. . . . . . . . . 10 |- F/ y w e. { y | ps }
21	19, 20	nfan 1988	. . . . . . . . 9 |- F/ y ( z = <. x , w >. /\ w e. { y | ps } )
22		nfv 1755	. . . . . . . . 9 |- F/ w ( z = <. x , y >. /\ ps )
23		opeq2 4126	. . . . . . . . . . 11 |- ( w = y -> <. x , w >. = <. x , y >. )
24	23	eqeq2d 2433	. . . . . . . . . 10 |- ( w = y -> ( z = <. x , w >. <-> z = <. x , y >. ) )
25		sbequ12 2052	. . . . . . . . . . . 12 |- ( y = w -> ( ps <-> [ w / y ] ps ) )
26	25	equcoms 1849	. . . . . . . . . . 11 |- ( w = y -> ( ps <-> [ w / y ] ps ) )
27		df-clab 2410	. . . . . . . . . . 11 |- ( w e. { y | ps } <-> [ w / y ] ps )
28	26, 27	syl6rbbr 267	. . . . . . . . . 10 |- ( w = y -> ( w e. { y | ps } <-> ps ) )
29	24, 28	anbi12d 715	. . . . . . . . 9 |- ( w = y -> ( ( z = <. x , w >. /\ w e. { y | ps } ) <-> ( z = <. x , y >. /\ ps ) ) )
30	21, 22, 29	cbvex 2082	. . . . . . . 8 |- ( E. w ( z = <. x , w >. /\ w e. { y | ps } ) <-> E. y ( z = <. x , y >. /\ ps ) )
31	4, 18, 30	3bitri 274	. . . . . . 7 |- ( z e. ( { x } X. { y | ps } ) <-> E. y ( z = <. x , y >. /\ ps ) )
32	31	anbi2i 698	. . . . . 6 |- ( ( x e. A /\ z e. ( { x } X. { y | ps } ) ) <-> ( x e. A /\ E. y ( z = <. x , y >. /\ ps ) ) )
33	1, 3, 32	3bitr4ri 281	. . . . 5 |- ( ( x e. A /\ z e. ( { x } X. { y | ps } ) ) <-> E. y ( z = <. x , y >. /\ ( x e. A /\ ps ) ) )
34	33	exbii 1712	. . . 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 4242	. . . . 5 |- ( z e. U_ x e. A ( { x } X. { y | ps } ) <-> E. x e. A z e. ( { x } X. { y | ps } ) )
36		df-rex 2715	. . . . 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 252	. . . 4 |- ( z e. U_ x e. A ( { x } X. { y | ps } ) <-> E. x ( x e. A /\ z e. ( { x } X. { y | ps } ) ) )
38		elopab 4666	. . . 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 280	. . 3 |- ( z e. U_ x e. A ( { x } X. { y | ps } ) <-> z e. { <. x , y >. | ( x e. A /\ ps ) } )
40	39	eqriv 2420	. 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 4600	. . . . 5 |- { x } e. _V
43		opabex3d.2	. . . . 5 |- ( ( ph /\ x e. A ) -> { y | ps } e. _V )
44		xpexg 6546	. . . . 5 |- ( ( { x } e. _V /\ { y | ps } e. _V ) -> ( { x } X. { y | ps } ) e. _V )
45	42, 43, 44	sylancr 667	. . . 4 |- ( ( ph /\ x e. A ) -> ( { x } X. { y | ps } ) e. _V )
46	45	ralrimiva 2774	. . 3 |- ( ph -> A. x e. A ( { x } X. { y | ps } ) e. _V )
47		iunexg 6722	. . 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 665	. 2 |- ( ph -> U_ x e. A ( { x } X. { y | ps } ) e. _V )
49	40, 48	syl5eqelr 2506	1 |- ( ph -> { <. x , y >. | ( x e. A /\ ps ) } e. _V )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   = wceq 1437   E.wex 1657   [wsb 1790   e. wcel 1872   {cab 2409   A.wral 2709   E.wrex 2710   _Vcvv 3017   {csn 3936   <.cop 3942   U_ciun 4237   {copab 4419   X. cxp 4789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-rep 4474  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-ral 2714  df-rex 2715  df-reu 2716  df-rab 2718  df-v 3019  df-sbc 3238  df-csb 3334  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-op 3943  df-uni 4158  df-iun 4239  df-br 4362  df-opab 4421  df-mpt 4422  df-id 4706  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-res 4803  df-ima 4804  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547

This theorem is referenced by:  wlks  25184  wlkres  25187  trls  25203  crcts  25287  cycls  25288  fpwrelmap  28263