Theorem dvdsrval 17489

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 5848	. . . . . . . . 9 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
3		dvdsr.1	. . . . . . . . 9 |- B = ( Base ` R )
4	2, 3	syl6eqr 2513	. . . . . . . 8 |- ( r = R -> ( Base ` r ) = B )
5	4	eleq2d 2524	. . . . . . 7 |- ( r = R -> ( x e. ( Base ` r ) <-> x e. B ) )
6	4	rexeqdv 3058	. . . . . . 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 708	. . . . . 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 5848	. . . . . . . . . . 11 |- ( r = R -> ( .r ` r ) = ( .r ` R ) )
9		dvdsr.3	. . . . . . . . . . 11 |- .x. = ( .r ` R )
10	8, 9	syl6eqr 2513	. . . . . . . . . 10 |- ( r = R -> ( .r ` r ) = .x. )
11	10	oveqd 6287	. . . . . . . . 9 |- ( r = R -> ( z ( .r ` r ) x ) = ( z .x. x ) )
12	11	eqeq1d 2456	. . . . . . . 8 |- ( r = R -> ( ( z ( .r ` r ) x ) = y <-> ( z .x. x ) = y ) )
13	12	rexbidv 2965	. . . . . . 7 |- ( r = R -> ( E. z e. B ( z ( .r ` r ) x ) = y <-> E. z e. B ( z .x. x ) = y ) )
14	13	anbi2d 701	. . . . . 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 4502	. . . 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 17485	. . . 4 |- ||r = ( r e. _V |-> { <. x , y >. | ( x e. ( Base ` r ) /\ E. z e. ( Base ` r ) ( z ( .r ` r ) x ) = y ) } )
18		fvex 5858	. . . . . 6 |- ( Base ` R ) e. _V
19	3, 18	eqeltri 2538	. . . . 5 |- B e. _V
20		eqcom 2463	. . . . . . . . 9 |- ( ( z .x. x ) = y <-> y = ( z .x. x ) )
21	20	rexbii 2956	. . . . . . . 8 |- ( E. z e. B ( z .x. x ) = y <-> E. z e. B y = ( z .x. x ) )
22	21	abbii 2588	. . . . . . 7 |- { y | E. z e. B ( z .x. x ) = y } = { y | E. z e. B y = ( z .x. x ) }
23	19	abrexex 6747	. . . . . . 7 |- { y | E. z e. B y = ( z .x. x ) } e. _V
24	22, 23	eqeltri 2538	. . . . . 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 6752	. . . 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 2507	. 2 |- ( R e. _V -> .|| = { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) } )
29		fvprc 5842	. . . 4 |- ( -. R e. _V -> ( ||r ` R ) = (/) )
30	1, 29	syl5eq 2507	. . 3 |- ( -. R e. _V -> .|| = (/) )
31		opabn0 4767	. . . . 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 3788	. . . . . . . 8 |- ( x e. B -> -. B = (/) )
33		fvprc 5842	. . . . . . . . 9 |- ( -. R e. _V -> ( Base ` R ) = (/) )
34	3, 33	syl5eq 2507	. . . . . . . 8 |- ( -. R e. _V -> B = (/) )
35	32, 34	nsyl2 127	. . . . . . 7 |- ( x e. B -> R e. _V )
36	35	adantr 463	. . . . . 6 |- ( ( x e. B /\ E. z e. B ( z .x. x ) = y ) -> R e. _V )
37	36	exlimivv 1728	. . . . 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 2687	. . 3 |- ( -. R e. _V -> { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) } = (/) )
40	30, 39	eqtr4d 2498	. 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 367   = wceq 1398   E.wex 1617   e. wcel 1823   {cab 2439   =/= wne 2649   E.wrex 2805   _Vcvv 3106   (/)c0 3783   {copab 4496   ` cfv 5570  (class class class)co 6270   Basecbs 14716   .rcmulr 14785   ||_rcdsr 17482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-dvdsr 17485

This theorem is referenced by:  dvdsr  17490  dvdsrpropd  17540  dvdsrzring  18696