Theorem opabex3 7146

Description: Existence of an ordered pair abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)

Hypotheses

Ref	Expression
opabex3.1	|- A e. _V
opabex3.2	|- ( x e. A -> { y | ph } e. _V )

Assertion

Ref	Expression
opabex3	|- { <. x , y >. | ( x e. A /\ ph ) } e. _V

Distinct variable group:   x, A, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem opabex3

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

Step	Hyp	Ref	Expression
1		19.42v 1918	. . . . . 6 |- ( E. y ( x e. A /\ ( z = <. x , y >. /\ ph ) ) <-> ( x e. A /\ E. y ( z = <. x , y >. /\ ph ) ) )
2		an12 838	. . . . . . 7 |- ( ( z = <. x , y >. /\ ( x e. A /\ ph ) ) <-> ( x e. A /\ ( z = <. x , y >. /\ ph ) ) )
3	2	exbii 1774	. . . . . 6 |- ( E. y ( z = <. x , y >. /\ ( x e. A /\ ph ) ) <-> E. y ( x e. A /\ ( z = <. x , y >. /\ ph ) ) )
4		elxp 5131	. . . . . . . 8 |- ( z e. ( { x } X. { y | ph } ) <-> E. v E. w ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ph } ) ) )
5		excom 2042	. . . . . . . . 9 |- ( E. v E. w ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ph } ) ) <-> E. w E. v ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ph } ) ) )
6		an12 838	. . . . . . . . . . . . 13 |- ( ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ph } ) ) <-> ( v e. { x } /\ ( z = <. v , w >. /\ w e. { y | ph } ) ) )
7		velsn 4193	. . . . . . . . . . . . . 14 |- ( v e. { x } <-> v = x )
8	7	anbi1i 731	. . . . . . . . . . . . 13 |- ( ( v e. { x } /\ ( z = <. v , w >. /\ w e. { y | ph } ) ) <-> ( v = x /\ ( z = <. v , w >. /\ w e. { y | ph } ) ) )
9	6, 8	bitri 264	. . . . . . . . . . . 12 |- ( ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ph } ) ) <-> ( v = x /\ ( z = <. v , w >. /\ w e. { y | ph } ) ) )
10	9	exbii 1774	. . . . . . . . . . 11 |- ( E. v ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ph } ) ) <-> E. v ( v = x /\ ( z = <. v , w >. /\ w e. { y | ph } ) ) )
11		vex 3203	. . . . . . . . . . . 12 |- x e. _V
12		opeq1 4402	. . . . . . . . . . . . . 14 |- ( v = x -> <. v , w >. = <. x , w >. )
13	12	eqeq2d 2632	. . . . . . . . . . . . 13 |- ( v = x -> ( z = <. v , w >. <-> z = <. x , w >. ) )
14	13	anbi1d 741	. . . . . . . . . . . 12 |- ( v = x -> ( ( z = <. v , w >. /\ w e. { y | ph } ) <-> ( z = <. x , w >. /\ w e. { y | ph } ) ) )
15	11, 14	ceqsexv 3242	. . . . . . . . . . 11 |- ( E. v ( v = x /\ ( z = <. v , w >. /\ w e. { y | ph } ) ) <-> ( z = <. x , w >. /\ w e. { y | ph } ) )
16	10, 15	bitri 264	. . . . . . . . . 10 |- ( E. v ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ph } ) ) <-> ( z = <. x , w >. /\ w e. { y | ph } ) )
17	16	exbii 1774	. . . . . . . . 9 |- ( E. w E. v ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ph } ) ) <-> E. w ( z = <. x , w >. /\ w e. { y | ph } ) )
18	5, 17	bitri 264	. . . . . . . 8 |- ( E. v E. w ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ph } ) ) <-> E. w ( z = <. x , w >. /\ w e. { y | ph } ) )
19		nfv 1843	. . . . . . . . . 10 |- F/ y z = <. x , w >.
20		nfsab1 2612	. . . . . . . . . 10 |- F/ y w e. { y | ph }
21	19, 20	nfan 1828	. . . . . . . . 9 |- F/ y ( z = <. x , w >. /\ w e. { y | ph } )
22		nfv 1843	. . . . . . . . 9 |- F/ w ( z = <. x , y >. /\ ph )
23		opeq2 4403	. . . . . . . . . . 11 |- ( w = y -> <. x , w >. = <. x , y >. )
24	23	eqeq2d 2632	. . . . . . . . . 10 |- ( w = y -> ( z = <. x , w >. <-> z = <. x , y >. ) )
25		sbequ12 2111	. . . . . . . . . . . 12 |- ( y = w -> ( ph <-> [ w / y ] ph ) )
26	25	equcoms 1947	. . . . . . . . . . 11 |- ( w = y -> ( ph <-> [ w / y ] ph ) )
27		df-clab 2609	. . . . . . . . . . 11 |- ( w e. { y | ph } <-> [ w / y ] ph )
28	26, 27	syl6rbbr 279	. . . . . . . . . 10 |- ( w = y -> ( w e. { y | ph } <-> ph ) )
29	24, 28	anbi12d 747	. . . . . . . . 9 |- ( w = y -> ( ( z = <. x , w >. /\ w e. { y | ph } ) <-> ( z = <. x , y >. /\ ph ) ) )
30	21, 22, 29	cbvex 2272	. . . . . . . 8 |- ( E. w ( z = <. x , w >. /\ w e. { y | ph } ) <-> E. y ( z = <. x , y >. /\ ph ) )
31	4, 18, 30	3bitri 286	. . . . . . 7 |- ( z e. ( { x } X. { y | ph } ) <-> E. y ( z = <. x , y >. /\ ph ) )
32	31	anbi2i 730	. . . . . 6 |- ( ( x e. A /\ z e. ( { x } X. { y | ph } ) ) <-> ( x e. A /\ E. y ( z = <. x , y >. /\ ph ) ) )
33	1, 3, 32	3bitr4ri 293	. . . . 5 |- ( ( x e. A /\ z e. ( { x } X. { y | ph } ) ) <-> E. y ( z = <. x , y >. /\ ( x e. A /\ ph ) ) )
34	33	exbii 1774	. . . 4 |- ( E. x ( x e. A /\ z e. ( { x } X. { y | ph } ) ) <-> E. x E. y ( z = <. x , y >. /\ ( x e. A /\ ph ) ) )
35		eliun 4524	. . . . 5 |- ( z e. U_ x e. A ( { x } X. { y | ph } ) <-> E. x e. A z e. ( { x } X. { y | ph } ) )
36		df-rex 2918	. . . . 5 |- ( E. x e. A z e. ( { x } X. { y | ph } ) <-> E. x ( x e. A /\ z e. ( { x } X. { y | ph } ) ) )
37	35, 36	bitri 264	. . . 4 |- ( z e. U_ x e. A ( { x } X. { y | ph } ) <-> E. x ( x e. A /\ z e. ( { x } X. { y | ph } ) ) )
38		elopab 4983	. . . 4 |- ( z e. { <. x , y >. | ( x e. A /\ ph ) } <-> E. x E. y ( z = <. x , y >. /\ ( x e. A /\ ph ) ) )
39	34, 37, 38	3bitr4i 292	. . 3 |- ( z e. U_ x e. A ( { x } X. { y | ph } ) <-> z e. { <. x , y >. | ( x e. A /\ ph ) } )
40	39	eqriv 2619	. 2 |- U_ x e. A ( { x } X. { y | ph } ) = { <. x , y >. | ( x e. A /\ ph ) }
41		opabex3.1	. . 3 |- A e. _V
42		snex 4908	. . . . 5 |- { x } e. _V
43		opabex3.2	. . . . 5 |- ( x e. A -> { y | ph } e. _V )
44		xpexg 6960	. . . . 5 |- ( ( { x } e. _V /\ { y | ph } e. _V ) -> ( { x } X. { y | ph } ) e. _V )
45	42, 43, 44	sylancr 695	. . . 4 |- ( x e. A -> ( { x } X. { y | ph } ) e. _V )
46	45	rgen 2922	. . 3 |- A. x e. A ( { x } X. { y | ph } ) e. _V
47		iunexg 7143	. . 3 |- ( ( A e. _V /\ A. x e. A ( { x } X. { y | ph } ) e. _V ) -> U_ x e. A ( { x } X. { y | ph } ) e. _V )
48	41, 46, 47	mp2an 708	. 2 |- U_ x e. A ( { x } X. { y | ph } ) e. _V
49	40, 48	eqeltrri 2698	1 |- { <. x , y >. | ( x e. A /\ ph ) } e. _V

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   [wsb 1880   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   _Vcvv 3200   {csn 4177   <.cop 4183   U_ciun 4520   {copab 4712   X. cxp 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by:  dvdsrval  18645  eulerpartlemgvv  30438