Theorem dvdsrval 16749

Description: Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-2015.)

Hypotheses

Ref	Expression
dvdsr.1	|- B = ( Base ` R )
dvdsr.2	|- .|| = ( ||r ` R )
dvdsr.3	|- .x. = ( .r ` R )

Assertion

Ref	Expression
dvdsrval	|- .|| = { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) }

Distinct variable groups:   x, y, .||   x, z, B, y   x, R, y, z   x, .x. , y, z

Allowed substitution hint:   .|| ( z)

Proof of Theorem dvdsrval

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvdsr.2	. . 3 |- .|| = ( ||r ` R )
2		fveq2 5703	. . . . . . . . 9 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
3		dvdsr.1	. . . . . . . . 9 |- B = ( Base ` R )
4	2, 3	syl6eqr 2493	. . . . . . . 8 |- ( r = R -> ( Base ` r ) = B )
5	4	eleq2d 2510	. . . . . . 7 |- ( r = R -> ( x e. ( Base ` r ) <-> x e. B ) )
6	4	rexeqdv 2936	. . . . . . 7 |- ( r = R -> ( E. z e. ( Base ` r ) ( z ( .r ` r ) x ) = y <-> E. z e. B ( z ( .r ` r ) x ) = y ) )
7	5, 6	anbi12d 710	. . . . . 6 |- ( r = R -> ( ( x e. ( Base ` r ) /\ E. z e. ( Base ` r ) ( z ( .r ` r ) x ) = y ) <-> ( x e. B /\ E. z e. B ( z ( .r ` r ) x ) = y ) ) )
8		fveq2 5703	. . . . . . . . . . 11 |- ( r = R -> ( .r ` r ) = ( .r ` R ) )
9		dvdsr.3	. . . . . . . . . . 11 |- .x. = ( .r ` R )
10	8, 9	syl6eqr 2493	. . . . . . . . . 10 |- ( r = R -> ( .r ` r ) = .x. )
11	10	oveqd 6120	. . . . . . . . 9 |- ( r = R -> ( z ( .r ` r ) x ) = ( z .x. x ) )
12	11	eqeq1d 2451	. . . . . . . 8 |- ( r = R -> ( ( z ( .r ` r ) x ) = y <-> ( z .x. x ) = y ) )
13	12	rexbidv 2748	. . . . . . 7 |- ( r = R -> ( E. z e. B ( z ( .r ` r ) x ) = y <-> E. z e. B ( z .x. x ) = y ) )
14	13	anbi2d 703	. . . . . 6 |- ( r = R -> ( ( x e. B /\ E. z e. B ( z ( .r ` r ) x ) = y ) <-> ( x e. B /\ E. z e. B ( z .x. x ) = y ) ) )
15	7, 14	bitrd 253	. . . . 5 |- ( r = R -> ( ( x e. ( Base ` r ) /\ E. z e. ( Base ` r ) ( z ( .r ` r ) x ) = y ) <-> ( x e. B /\ E. z e. B ( z .x. x ) = y ) ) )
16	15	opabbidv 4367	. . . 4 |- ( r = R -> { <. x , y >. | ( x e. ( Base ` r ) /\ E. z e. ( Base ` r ) ( z ( .r ` r ) x ) = y ) } = { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) } )
17		df-dvdsr 16745	. . . 4 |- ||r = ( r e. _V |-> { <. x , y >. | ( x e. ( Base ` r ) /\ E. z e. ( Base ` r ) ( z ( .r ` r ) x ) = y ) } )
18		fvex 5713	. . . . . 6 |- ( Base ` R ) e. _V
19	3, 18	eqeltri 2513	. . . . 5 |- B e. _V
20		eqcom 2445	. . . . . . . . 9 |- ( ( z .x. x ) = y <-> y = ( z .x. x ) )
21	20	rexbii 2752	. . . . . . . 8 |- ( E. z e. B ( z .x. x ) = y <-> E. z e. B y = ( z .x. x ) )
22	21	abbii 2561	. . . . . . 7 |- { y | E. z e. B ( z .x. x ) = y } = { y | E. z e. B y = ( z .x. x ) }
23	19	abrexex 6563	. . . . . . 7 |- { y | E. z e. B y = ( z .x. x ) } e. _V
24	22, 23	eqeltri 2513	. . . . . 6 |- { y | E. z e. B ( z .x. x ) = y } e. _V
25	24	a1i 11	. . . . 5 |- ( x e. B -> { y | E. z e. B ( z .x. x ) = y } e. _V )
26	19, 25	opabex3 6568	. . . 4 |- { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) } e. _V
27	16, 17, 26	fvmpt 5786	. . 3 |- ( R e. _V -> ( ||r ` R ) = { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) } )
28	1, 27	syl5eq 2487	. 2 |- ( R e. _V -> .|| = { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) } )
29		fvprc 5697	. . . 4 |- ( -. R e. _V -> ( ||r ` R ) = (/) )
30	1, 29	syl5eq 2487	. . 3 |- ( -. R e. _V -> .|| = (/) )
31		opabn0 4631	. . . . 5 |- ( { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) } =/= (/) <-> E. x E. y ( x e. B /\ E. z e. B ( z .x. x ) = y ) )
32		n0i 3654	. . . . . . . 8 |- ( x e. B -> -. B = (/) )
33		fvprc 5697	. . . . . . . . 9 |- ( -. R e. _V -> ( Base ` R ) = (/) )
34	3, 33	syl5eq 2487	. . . . . . . 8 |- ( -. R e. _V -> B = (/) )
35	32, 34	nsyl2 127	. . . . . . 7 |- ( x e. B -> R e. _V )
36	35	adantr 465	. . . . . 6 |- ( ( x e. B /\ E. z e. B ( z .x. x ) = y ) -> R e. _V )
37	36	exlimivv 1689	. . . . 5 |- ( E. x E. y ( x e. B /\ E. z e. B ( z .x. x ) = y ) -> R e. _V )
38	31, 37	sylbi 195	. . . 4 |- ( { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) } =/= (/) -> R e. _V )
39	38	necon1bi 2666	. . 3 |- ( -. R e. _V -> { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) } = (/) )
40	30, 39	eqtr4d 2478	. 2 |- ( -. R e. _V -> .|| = { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) } )
41	28, 40	pm2.61i 164	1 |- .|| = { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) }

Syntax hints:   -. wn 3   /\ wa 369   = wceq 1369   E.wex 1586   e. wcel 1756   {cab 2429   =/= wne 2618   E.wrex 2728   _Vcvv 2984   (/)c0 3649   {copab 4361   ` cfv 5430  (class class class)co 6103   Basecbs 14186   .rcmulr 14251   ||_rcdsr 16742

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 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384

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 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-ov 6106  df-dvdsr 16745

This theorem is referenced by:  dvdsr  16750  dvdsrpropd  16800  dvdsrzring  17913  dvdsrz  17914