Theorem elxp5 5190

Description: Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 5189 when the double intersection does not create class existence problems (caused by int0 3913). (Contributed by NM, 1-Aug-2004.)

Assertion

Ref	Expression
elxp5	|- ( A e. ( B X. C ) <-> ( A = <. |^| |^| A , U. ran { A } >. /\ ( |^| |^| A e. B /\ U. ran { A } e. C ) ) )

Proof of Theorem elxp5

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2788	. 2 |- ( A e. ( B X. C ) -> A e. _V )
2		elex 2788	. . . 4 |- ( |^| |^| A e. B -> |^| |^| A e. _V )
3		elex 2788	. . . 4 |- ( U. ran { A } e. C -> U. ran { A } e. _V )
4	2, 3	anim12i 338	. . 3 |- ( ( |^| |^| A e. B /\ U. ran { A } e. C ) -> ( |^| |^| A e. _V /\ U. ran { A } e. _V ) )
5		opexg 4290	. . . . 5 |- ( ( |^| |^| A e. _V /\ U. ran { A } e. _V ) -> <. |^| |^| A , U. ran { A } >. e. _V )
6	5	adantl 277	. . . 4 |- ( ( A = <. |^| |^| A , U. ran { A } >. /\ ( |^| |^| A e. _V /\ U. ran { A } e. _V ) ) -> <. |^| |^| A , U. ran { A } >. e. _V )
7		eleq1 2270	. . . . 5 |- ( A = <. |^| |^| A , U. ran { A } >. -> ( A e. _V <-> <. |^| |^| A , U. ran { A } >. e. _V ) )
8	7	adantr 276	. . . 4 |- ( ( A = <. |^| |^| A , U. ran { A } >. /\ ( |^| |^| A e. _V /\ U. ran { A } e. _V ) ) -> ( A e. _V <-> <. |^| |^| A , U. ran { A } >. e. _V ) )
9	6, 8	mpbird 167	. . 3 |- ( ( A = <. |^| |^| A , U. ran { A } >. /\ ( |^| |^| A e. _V /\ U. ran { A } e. _V ) ) -> A e. _V )
10	4, 9	sylan2 286	. 2 |- ( ( A = <. |^| |^| A , U. ran { A } >. /\ ( |^| |^| A e. B /\ U. ran { A } e. C ) ) -> A e. _V )
11		elxp 4710	. . . 4 |- ( A e. ( B X. C ) <-> E. x E. y ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) )
12		sneq 3654	. . . . . . . . . . . . . 14 |- ( A = <. x , y >. -> { A } = { <. x , y >. } )
13	12	rneqd 4926	. . . . . . . . . . . . 13 |- ( A = <. x , y >. -> ran { A } = ran { <. x , y >. } )
14	13	unieqd 3875	. . . . . . . . . . . 12 |- ( A = <. x , y >. -> U. ran { A } = U. ran { <. x , y >. } )
15		vex 2779	. . . . . . . . . . . . 13 |- x e. _V
16		vex 2779	. . . . . . . . . . . . 13 |- y e. _V
17	15, 16	op2nda 5186	. . . . . . . . . . . 12 |- U. ran { <. x , y >. } = y
18	14, 17	eqtr2di 2257	. . . . . . . . . . 11 |- ( A = <. x , y >. -> y = U. ran { A } )
19	18	pm4.71ri 392	. . . . . . . . . 10 |- ( A = <. x , y >. <-> ( y = U. ran { A } /\ A = <. x , y >. ) )
20	19	anbi1i 458	. . . . . . . . 9 |- ( ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) <-> ( ( y = U. ran { A } /\ A = <. x , y >. ) /\ ( x e. B /\ y e. C ) ) )
21		anass 401	. . . . . . . . 9 |- ( ( ( y = U. ran { A } /\ A = <. x , y >. ) /\ ( x e. B /\ y e. C ) ) <-> ( y = U. ran { A } /\ ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) ) )
22	20, 21	bitri 184	. . . . . . . 8 |- ( ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) <-> ( y = U. ran { A } /\ ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) ) )
23	22	exbii 1629	. . . . . . 7 |- ( E. y ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) <-> E. y ( y = U. ran { A } /\ ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) ) )
24		snexg 4244	. . . . . . . . . 10 |- ( A e. _V -> { A } e. _V )
25		rnexg 4962	. . . . . . . . . 10 |- ( { A } e. _V -> ran { A } e. _V )
26	24, 25	syl 14	. . . . . . . . 9 |- ( A e. _V -> ran { A } e. _V )
27		uniexg 4504	. . . . . . . . 9 |- ( ran { A } e. _V -> U. ran { A } e. _V )
28	26, 27	syl 14	. . . . . . . 8 |- ( A e. _V -> U. ran { A } e. _V )
29		opeq2 3834	. . . . . . . . . . 11 |- ( y = U. ran { A } -> <. x , y >. = <. x , U. ran { A } >. )
30	29	eqeq2d 2219	. . . . . . . . . 10 |- ( y = U. ran { A } -> ( A = <. x , y >. <-> A = <. x , U. ran { A } >. ) )
31		eleq1 2270	. . . . . . . . . . 11 |- ( y = U. ran { A } -> ( y e. C <-> U. ran { A } e. C ) )
32	31	anbi2d 464	. . . . . . . . . 10 |- ( y = U. ran { A } -> ( ( x e. B /\ y e. C ) <-> ( x e. B /\ U. ran { A } e. C ) ) )
33	30, 32	anbi12d 473	. . . . . . . . 9 |- ( y = U. ran { A } -> ( ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) <-> ( A = <. x , U. ran { A } >. /\ ( x e. B /\ U. ran { A } e. C ) ) ) )
34	33	ceqsexgv 2909	. . . . . . . 8 |- ( U. ran { A } e. _V -> ( E. y ( y = U. ran { A } /\ ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) ) <-> ( A = <. x , U. ran { A } >. /\ ( x e. B /\ U. ran { A } e. C ) ) ) )
35	28, 34	syl 14	. . . . . . 7 |- ( A e. _V -> ( E. y ( y = U. ran { A } /\ ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) ) <-> ( A = <. x , U. ran { A } >. /\ ( x e. B /\ U. ran { A } e. C ) ) ) )
36	23, 35	bitrid 192	. . . . . 6 |- ( A e. _V -> ( E. y ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) <-> ( A = <. x , U. ran { A } >. /\ ( x e. B /\ U. ran { A } e. C ) ) ) )
37		inteq 3902	. . . . . . . . . . . 12 |- ( A = <. x , U. ran { A } >. -> |^| A = |^| <. x , U. ran { A } >. )
38	37	inteqd 3904	. . . . . . . . . . 11 |- ( A = <. x , U. ran { A } >. -> |^| |^| A = |^| |^| <. x , U. ran { A } >. )
39	38	adantl 277	. . . . . . . . . 10 |- ( ( A e. _V /\ A = <. x , U. ran { A } >. ) -> |^| |^| A = |^| |^| <. x , U. ran { A } >. )
40		op1stbg 4544	. . . . . . . . . . . 12 |- ( ( x e. _V /\ U. ran { A } e. _V ) -> |^| |^| <. x , U. ran { A } >. = x )
41	15, 28, 40	sylancr 414	. . . . . . . . . . 11 |- ( A e. _V -> |^| |^| <. x , U. ran { A } >. = x )
42	41	adantr 276	. . . . . . . . . 10 |- ( ( A e. _V /\ A = <. x , U. ran { A } >. ) -> |^| |^| <. x , U. ran { A } >. = x )
43	39, 42	eqtr2d 2241	. . . . . . . . 9 |- ( ( A e. _V /\ A = <. x , U. ran { A } >. ) -> x = |^| |^| A )
44	43	ex 115	. . . . . . . 8 |- ( A e. _V -> ( A = <. x , U. ran { A } >. -> x = |^| |^| A ) )
45	44	pm4.71rd 394	. . . . . . 7 |- ( A e. _V -> ( A = <. x , U. ran { A } >. <-> ( x = |^| |^| A /\ A = <. x , U. ran { A } >. ) ) )
46	45	anbi1d 465	. . . . . 6 |- ( A e. _V -> ( ( A = <. x , U. ran { A } >. /\ ( x e. B /\ U. ran { A } e. C ) ) <-> ( ( x = |^| |^| A /\ A = <. x , U. ran { A } >. ) /\ ( x e. B /\ U. ran { A } e. C ) ) ) )
47		anass 401	. . . . . . 7 |- ( ( ( x = |^| |^| A /\ A = <. x , U. ran { A } >. ) /\ ( x e. B /\ U. ran { A } e. C ) ) <-> ( x = |^| |^| A /\ ( A = <. x , U. ran { A } >. /\ ( x e. B /\ U. ran { A } e. C ) ) ) )
48	47	a1i 9	. . . . . 6 |- ( A e. _V -> ( ( ( x = |^| |^| A /\ A = <. x , U. ran { A } >. ) /\ ( x e. B /\ U. ran { A } e. C ) ) <-> ( x = |^| |^| A /\ ( A = <. x , U. ran { A } >. /\ ( x e. B /\ U. ran { A } e. C ) ) ) ) )
49	36, 46, 48	3bitrd 214	. . . . 5 |- ( A e. _V -> ( E. y ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) <-> ( x = |^| |^| A /\ ( A = <. x , U. ran { A } >. /\ ( x e. B /\ U. ran { A } e. C ) ) ) ) )
50	49	exbidv 1849	. . . 4 |- ( A e. _V -> ( E. x E. y ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) <-> E. x ( x = |^| |^| A /\ ( A = <. x , U. ran { A } >. /\ ( x e. B /\ U. ran { A } e. C ) ) ) ) )
51	11, 50	bitrid 192	. . 3 |- ( A e. _V -> ( A e. ( B X. C ) <-> E. x ( x = |^| |^| A /\ ( A = <. x , U. ran { A } >. /\ ( x e. B /\ U. ran { A } e. C ) ) ) ) )
52		eqvisset 2787	. . . . . 6 |- ( x = |^| |^| A -> |^| |^| A e. _V )
53	52	adantr 276	. . . . 5 |- ( ( x = |^| |^| A /\ ( A = <. x , U. ran { A } >. /\ ( x e. B /\ U. ran { A } e. C ) ) ) -> |^| |^| A e. _V )
54	53	exlimiv 1622	. . . 4 |- ( E. x ( x = |^| |^| A /\ ( A = <. x , U. ran { A } >. /\ ( x e. B /\ U. ran { A } e. C ) ) ) -> |^| |^| A e. _V )
55	2	ad2antrl 490	. . . 4 |- ( ( A = <. |^| |^| A , U. ran { A } >. /\ ( |^| |^| A e. B /\ U. ran { A } e. C ) ) -> |^| |^| A e. _V )
56		opeq1 3833	. . . . . . 7 |- ( x = |^| |^| A -> <. x , U. ran { A } >. = <. |^| |^| A , U. ran { A } >. )
57	56	eqeq2d 2219	. . . . . 6 |- ( x = |^| |^| A -> ( A = <. x , U. ran { A } >. <-> A = <. |^| |^| A , U. ran { A } >. ) )
58		eleq1 2270	. . . . . . 7 |- ( x = |^| |^| A -> ( x e. B <-> |^| |^| A e. B ) )
59	58	anbi1d 465	. . . . . 6 |- ( x = |^| |^| A -> ( ( x e. B /\ U. ran { A } e. C ) <-> ( |^| |^| A e. B /\ U. ran { A } e. C ) ) )
60	57, 59	anbi12d 473	. . . . 5 |- ( x = |^| |^| A -> ( ( A = <. x , U. ran { A } >. /\ ( x e. B /\ U. ran { A } e. C ) ) <-> ( A = <. |^| |^| A , U. ran { A } >. /\ ( |^| |^| A e. B /\ U. ran { A } e. C ) ) ) )
61	60	ceqsexgv 2909	. . . 4 |- ( |^| |^| A e. _V -> ( E. x ( x = |^| |^| A /\ ( A = <. x , U. ran { A } >. /\ ( x e. B /\ U. ran { A } e. C ) ) ) <-> ( A = <. |^| |^| A , U. ran { A } >. /\ ( |^| |^| A e. B /\ U. ran { A } e. C ) ) ) )
62	54, 55, 61	pm5.21nii 706	. . 3 |- ( E. x ( x = |^| |^| A /\ ( A = <. x , U. ran { A } >. /\ ( x e. B /\ U. ran { A } e. C ) ) ) <-> ( A = <. |^| |^| A , U. ran { A } >. /\ ( |^| |^| A e. B /\ U. ran { A } e. C ) ) )
63	51, 62	bitrdi 196	. 2 |- ( A e. _V -> ( A e. ( B X. C ) <-> ( A = <. |^| |^| A , U. ran { A } >. /\ ( |^| |^| A e. B /\ U. ran { A } e. C ) ) ) )
64	1, 10, 63	pm5.21nii 706	1 |- ( A e. ( B X. C ) <-> ( A = <. |^| |^| A , U. ran { A } >. /\ ( |^| |^| A e. B /\ U. ran { A } e. C ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1373   E.wex 1516   e. wcel 2178   _Vcvv 2776   {csn 3643   <.cop 3646   U.cuni 3864   |^|cint 3899   X. cxp 4691   ran crn 4694

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-xp 4699  df-rel 4700  df-cnv 4701  df-dm 4703  df-rn 4704

This theorem is referenced by: (None)