Theorem opabex3 6341

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 1962	. . . . . 6 |- ( E. y ( x e. A /\ ( z = <. x , y >. /\ ph ) ) <-> ( x e. A /\ E. y ( z = <. x , y >. /\ ph ) ) )
2		an12 567	. . . . . . 7 |- ( ( z = <. x , y >. /\ ( x e. A /\ ph ) ) <-> ( x e. A /\ ( z = <. x , y >. /\ ph ) ) )
3	2	exbii 1658	. . . . . 6 |- ( E. y ( z = <. x , y >. /\ ( x e. A /\ ph ) ) <-> E. y ( x e. A /\ ( z = <. x , y >. /\ ph ) ) )
4		elxp 4786	. . . . . . . 8 |- ( z e. ( { x } X. { y | ph } ) <-> E. v E. w ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ph } ) ) )
5		excom 1716	. . . . . . . . 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 567	. . . . . . . . . . . . 13 |- ( ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ph } ) ) <-> ( v e. { x } /\ ( z = <. v , w >. /\ w e. { y | ph } ) ) )
7		velsn 3722	. . . . . . . . . . . . . 14 |- ( v e. { x } <-> v = x )
8	7	anbi1i 462	. . . . . . . . . . . . 13 |- ( ( v e. { x } /\ ( z = <. v , w >. /\ w e. { y | ph } ) ) <-> ( v = x /\ ( z = <. v , w >. /\ w e. { y | ph } ) ) )
9	6, 8	bitri 184	. . . . . . . . . . . 12 |- ( ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ph } ) ) <-> ( v = x /\ ( z = <. v , w >. /\ w e. { y | ph } ) ) )
10	9	exbii 1658	. . . . . . . . . . 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 2824	. . . . . . . . . . . 12 |- x e. _V
12		opeq1 3899	. . . . . . . . . . . . . 14 |- ( v = x -> <. v , w >. = <. x , w >. )
13	12	eqeq2d 2250	. . . . . . . . . . . . 13 |- ( v = x -> ( z = <. v , w >. <-> z = <. x , w >. ) )
14	13	anbi1d 469	. . . . . . . . . . . 12 |- ( v = x -> ( ( z = <. v , w >. /\ w e. { y | ph } ) <-> ( z = <. x , w >. /\ w e. { y | ph } ) ) )
15	11, 14	ceqsexv 2861	. . . . . . . . . . 11 |- ( E. v ( v = x /\ ( z = <. v , w >. /\ w e. { y | ph } ) ) <-> ( z = <. x , w >. /\ w e. { y | ph } ) )
16	10, 15	bitri 184	. . . . . . . . . 10 |- ( E. v ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ph } ) ) <-> ( z = <. x , w >. /\ w e. { y | ph } ) )
17	16	exbii 1658	. . . . . . . . 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 184	. . . . . . . 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 1581	. . . . . . . . . 10 |- F/ y z = <. x , w >.
20		nfsab1 2228	. . . . . . . . . 10 |- F/ y w e. { y | ph }
21	19, 20	nfan 1618	. . . . . . . . 9 |- F/ y ( z = <. x , w >. /\ w e. { y | ph } )
22		nfv 1581	. . . . . . . . 9 |- F/ w ( z = <. x , y >. /\ ph )
23		opeq2 3900	. . . . . . . . . . 11 |- ( w = y -> <. x , w >. = <. x , y >. )
24	23	eqeq2d 2250	. . . . . . . . . 10 |- ( w = y -> ( z = <. x , w >. <-> z = <. x , y >. ) )
25		df-clab 2225	. . . . . . . . . . 11 |- ( w e. { y | ph } <-> [ w / y ] ph )
26		sbequ12 1824	. . . . . . . . . . . 12 |- ( y = w -> ( ph <-> [ w / y ] ph ) )
27	26	equcoms 1760	. . . . . . . . . . 11 |- ( w = y -> ( ph <-> [ w / y ] ph ) )
28	25, 27	bitr4id 199	. . . . . . . . . 10 |- ( w = y -> ( w e. { y | ph } <-> ph ) )
29	24, 28	anbi12d 477	. . . . . . . . 9 |- ( w = y -> ( ( z = <. x , w >. /\ w e. { y | ph } ) <-> ( z = <. x , y >. /\ ph ) ) )
30	21, 22, 29	cbvex 1809	. . . . . . . 8 |- ( E. w ( z = <. x , w >. /\ w e. { y | ph } ) <-> E. y ( z = <. x , y >. /\ ph ) )
31	4, 18, 30	3bitri 206	. . . . . . 7 |- ( z e. ( { x } X. { y | ph } ) <-> E. y ( z = <. x , y >. /\ ph ) )
32	31	anbi2i 461	. . . . . 6 |- ( ( x e. A /\ z e. ( { x } X. { y | ph } ) ) <-> ( x e. A /\ E. y ( z = <. x , y >. /\ ph ) ) )
33	1, 3, 32	3bitr4ri 213	. . . . 5 |- ( ( x e. A /\ z e. ( { x } X. { y | ph } ) ) <-> E. y ( z = <. x , y >. /\ ( x e. A /\ ph ) ) )
34	33	exbii 1658	. . . 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 4011	. . . . 5 |- ( z e. U_ x e. A ( { x } X. { y | ph } ) <-> E. x e. A z e. ( { x } X. { y | ph } ) )
36		df-rex 2534	. . . . 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 184	. . . 4 |- ( z e. U_ x e. A ( { x } X. { y | ph } ) <-> E. x ( x e. A /\ z e. ( { x } X. { y | ph } ) ) )
38		elopab 4395	. . . 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 212	. . 3 |- ( z e. U_ x e. A ( { x } X. { y | ph } ) <-> z e. { <. x , y >. | ( x e. A /\ ph ) } )
40	39	eqriv 2235	. 2 |- U_ x e. A ( { x } X. { y | ph } ) = { <. x , y >. | ( x e. A /\ ph ) }
41		opabex3.1	. . 3 |- A e. _V
42		snexg 4316	. . . . . 6 |- ( x e. _V -> { x } e. _V )
43	11, 42	ax-mp 5	. . . . 5 |- { x } e. _V
44		opabex3.2	. . . . 5 |- ( x e. A -> { y | ph } e. _V )
45		xpexg 4884	. . . . 5 |- ( ( { x } e. _V /\ { y | ph } e. _V ) -> ( { x } X. { y | ph } ) e. _V )
46	43, 44, 45	sylancr 418	. . . 4 |- ( x e. A -> ( { x } X. { y | ph } ) e. _V )
47	46	rgen 2603	. . 3 |- A. x e. A ( { x } X. { y | ph } ) e. _V
48		iunexg 6338	. . 3 |- ( ( A e. _V /\ A. x e. A ( { x } X. { y | ph } ) e. _V ) -> U_ x e. A ( { x } X. { y | ph } ) e. _V )
49	41, 47, 48	mp2an 430	. 2 |- U_ x e. A ( { x } X. { y | ph } ) e. _V
50	40, 49	eqeltrri 2312	1 |- { <. x , y >. | ( x e. A /\ ph ) } e. _V

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   E.wex 1545   [wsb 1815   e. wcel 2209   {cab 2224   A.wral 2528   E.wrex 2529   _Vcvv 2821   {csn 3705   <.cop 3708   U_ciun 4007   {copab 4186   X. cxp 4767

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380

This theorem is referenced by: (None)