Theorem dvdsrval 17883

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 5847	. . . . . . . . 9 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
3		dvdsr.1	. . . . . . . . 9 |- B = ( Base ` R )
4	2, 3	syl6eqr 2503	. . . . . . . 8 |- ( r = R -> ( Base ` r ) = B )
5	4	eleq2d 2514	. . . . . . 7 |- ( r = R -> ( x e. ( Base ` r ) <-> x e. B ) )
6	4	rexeqdv 2961	. . . . . . 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 722	. . . . . 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 5847	. . . . . . . . . . 11 |- ( r = R -> ( .r ` r ) = ( .r ` R ) )
9		dvdsr.3	. . . . . . . . . . 11 |- .x. = ( .r ` R )
10	8, 9	syl6eqr 2503	. . . . . . . . . 10 |- ( r = R -> ( .r ` r ) = .x. )
11	10	oveqd 6292	. . . . . . . . 9 |- ( r = R -> ( z ( .r ` r ) x ) = ( z .x. x ) )
12	11	eqeq1d 2453	. . . . . . . 8 |- ( r = R -> ( ( z ( .r ` r ) x ) = y <-> ( z .x. x ) = y ) )
13	12	rexbidv 2872	. . . . . . 7 |- ( r = R -> ( E. z e. B ( z ( .r ` r ) x ) = y <-> E. z e. B ( z .x. x ) = y ) )
14	13	anbi2d 715	. . . . . 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 261	. . . . 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 4437	. . . 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 17879	. . . 4 |- ||r = ( r e. _V |-> { <. x , y >. | ( x e. ( Base ` r ) /\ E. z e. ( Base ` r ) ( z ( .r ` r ) x ) = y ) } )
18		fvex 5857	. . . . . 6 |- ( Base ` R ) e. _V
19	3, 18	eqeltri 2525	. . . . 5 |- B e. _V
20		eqcom 2458	. . . . . . . . 9 |- ( ( z .x. x ) = y <-> y = ( z .x. x ) )
21	20	rexbii 2861	. . . . . . . 8 |- ( E. z e. B ( z .x. x ) = y <-> E. z e. B y = ( z .x. x ) )
22	21	abbii 2567	. . . . . . 7 |- { y | E. z e. B ( z .x. x ) = y } = { y | E. z e. B y = ( z .x. x ) }
23	19	abrexex 6754	. . . . . . 7 |- { y | E. z e. B y = ( z .x. x ) } e. _V
24	22, 23	eqeltri 2525	. . . . . 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 6759	. . . 4 |- { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) } e. _V
27	16, 17, 26	fvmpt 5931	. . 3 |- ( R e. _V -> ( ||r ` R ) = { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) } )
28	1, 27	syl5eq 2497	. 2 |- ( R e. _V -> .|| = { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) } )
29		fvprc 5841	. . . 4 |- ( -. R e. _V -> ( ||r ` R ) = (/) )
30	1, 29	syl5eq 2497	. . 3 |- ( -. R e. _V -> .|| = (/) )
31		opabn0 4704	. . . . 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 3703	. . . . . . . 8 |- ( x e. B -> -. B = (/) )
33		fvprc 5841	. . . . . . . . 9 |- ( -. R e. _V -> ( Base ` R ) = (/) )
34	3, 33	syl5eq 2497	. . . . . . . 8 |- ( -. R e. _V -> B = (/) )
35	32, 34	nsyl2 132	. . . . . . 7 |- ( x e. B -> R e. _V )
36	35	adantr 471	. . . . . 6 |- ( ( x e. B /\ E. z e. B ( z .x. x ) = y ) -> R e. _V )
37	36	exlimivv 1781	. . . . 5 |- ( E. x E. y ( x e. B /\ E. z e. B ( z .x. x ) = y ) -> R e. _V )
38	31, 37	sylbi 200	. . . 4 |- ( { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) } =/= (/) -> R e. _V )
39	38	necon1bi 2651	. . 3 |- ( -. R e. _V -> { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) } = (/) )
40	30, 39	eqtr4d 2488	. 2 |- ( -. R e. _V -> .|| = { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) } )
41	28, 40	pm2.61i 169	1 |- .|| = { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) }

Syntax hints:   -. wn 3   /\ wa 375   = wceq 1447   E.wex 1666   e. wcel 1890   {cab 2437   =/= wne 2621   E.wrex 2737   _Vcvv 3012   (/)c0 3698   {copab 4431   ` cfv 5560  (class class class)co 6275   Basecbs 15131   .rcmulr 15201   ||_rcdsr 17876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-8 1892  ax-9 1899  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431  ax-rep 4486  ax-sep 4496  ax-nul 4505  ax-pow 4553  ax-pr 4611  ax-un 6570

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1450  df-ex 1667  df-nf 1671  df-sb 1801  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3014  df-sbc 3235  df-csb 3331  df-dif 3374  df-un 3376  df-in 3378  df-ss 3385  df-nul 3699  df-if 3849  df-pw 3920  df-sn 3936  df-pr 3938  df-op 3942  df-uni 4168  df-iun 4249  df-br 4374  df-opab 4433  df-mpt 4434  df-id 4726  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5524  df-fun 5562  df-fn 5563  df-f 5564  df-f1 5565  df-fo 5566  df-f1o 5567  df-fv 5568  df-ov 6278  df-dvdsr 17879

This theorem is referenced by:  dvdsr  17884  dvdsrpropd  17934  dvdsrzring  19062