Theorem dvdsrval 17107

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 5866	. . . . . . . . 9 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
3		dvdsr.1	. . . . . . . . 9 |- B = ( Base ` R )
4	2, 3	syl6eqr 2526	. . . . . . . 8 |- ( r = R -> ( Base ` r ) = B )
5	4	eleq2d 2537	. . . . . . 7 |- ( r = R -> ( x e. ( Base ` r ) <-> x e. B ) )
6	4	rexeqdv 3065	. . . . . . 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 5866	. . . . . . . . . . 11 |- ( r = R -> ( .r ` r ) = ( .r ` R ) )
9		dvdsr.3	. . . . . . . . . . 11 |- .x. = ( .r ` R )
10	8, 9	syl6eqr 2526	. . . . . . . . . 10 |- ( r = R -> ( .r ` r ) = .x. )
11	10	oveqd 6302	. . . . . . . . 9 |- ( r = R -> ( z ( .r ` r ) x ) = ( z .x. x ) )
12	11	eqeq1d 2469	. . . . . . . 8 |- ( r = R -> ( ( z ( .r ` r ) x ) = y <-> ( z .x. x ) = y ) )
13	12	rexbidv 2973	. . . . . . 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 4510	. . . 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 17103	. . . 4 |- ||r = ( r e. _V |-> { <. x , y >. | ( x e. ( Base ` r ) /\ E. z e. ( Base ` r ) ( z ( .r ` r ) x ) = y ) } )
18		fvex 5876	. . . . . 6 |- ( Base ` R ) e. _V
19	3, 18	eqeltri 2551	. . . . 5 |- B e. _V
20		eqcom 2476	. . . . . . . . 9 |- ( ( z .x. x ) = y <-> y = ( z .x. x ) )
21	20	rexbii 2965	. . . . . . . 8 |- ( E. z e. B ( z .x. x ) = y <-> E. z e. B y = ( z .x. x ) )
22	21	abbii 2601	. . . . . . 7 |- { y | E. z e. B ( z .x. x ) = y } = { y | E. z e. B y = ( z .x. x ) }
23	19	abrexex 6759	. . . . . . 7 |- { y | E. z e. B y = ( z .x. x ) } e. _V
24	22, 23	eqeltri 2551	. . . . . 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 6764	. . . 4 |- { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) } e. _V
27	16, 17, 26	fvmpt 5951	. . 3 |- ( R e. _V -> ( ||r ` R ) = { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) } )
28	1, 27	syl5eq 2520	. 2 |- ( R e. _V -> .|| = { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) } )
29		fvprc 5860	. . . 4 |- ( -. R e. _V -> ( ||r ` R ) = (/) )
30	1, 29	syl5eq 2520	. . 3 |- ( -. R e. _V -> .|| = (/) )
31		opabn0 4778	. . . . 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 3790	. . . . . . . 8 |- ( x e. B -> -. B = (/) )
33		fvprc 5860	. . . . . . . . 9 |- ( -. R e. _V -> ( Base ` R ) = (/) )
34	3, 33	syl5eq 2520	. . . . . . . 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 1699	. . . . 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 2700	. . 3 |- ( -. R e. _V -> { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) } = (/) )
40	30, 39	eqtr4d 2511	. 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 1379   E.wex 1596   e. wcel 1767   {cab 2452   =/= wne 2662   E.wrex 2815   _Vcvv 3113   (/)c0 3785   {copab 4504   ` cfv 5588  (class class class)co 6285   Basecbs 14493   .rcmulr 14559   ||_rcdsr 17100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-dvdsr 17103

This theorem is referenced by:  dvdsr  17108  dvdsrpropd  17158  dvdsrzring  18314  dvdsrz  18315