Theorem reldm 5832

Description: An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.)

Assertion

Ref	Expression
reldm	|- ( Rel A -> dom A = ran ( x e. A |-> ( 1st ` x ) ) )

Distinct variable group:   x, A

Proof of Theorem reldm

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

Step	Hyp	Ref	Expression
1		releldm2 5831	. . 3 |- ( Rel A -> ( y e. dom A <-> E. z e. A ( 1st ` z ) = y ) )
2		vex 2604	. . . . . . 7 |- x e. _V
3		1stexg 5814	. . . . . . 7 |- ( x e. _V -> ( 1st ` x ) e. _V )
4	2, 3	ax-mp 7	. . . . . 6 |- ( 1st ` x ) e. _V
5		eqid 2081	. . . . . 6 |- ( x e. A |-> ( 1st ` x ) ) = ( x e. A |-> ( 1st ` x ) )
6	4, 5	fnmpti 5047	. . . . 5 |- ( x e. A |-> ( 1st ` x ) ) Fn A
7		fvelrnb 5242	. . . . 5 |- ( ( x e. A |-> ( 1st ` x ) ) Fn A -> ( y e. ran ( x e. A |-> ( 1st ` x ) ) <-> E. z e. A ( ( x e. A |-> ( 1st ` x ) ) ` z ) = y ) )
8	6, 7	ax-mp 7	. . . 4 |- ( y e. ran ( x e. A |-> ( 1st ` x ) ) <-> E. z e. A ( ( x e. A |-> ( 1st ` x ) ) ` z ) = y )
9		fveq2 5198	. . . . . . . 8 |- ( x = z -> ( 1st ` x ) = ( 1st ` z ) )
10		vex 2604	. . . . . . . . 9 |- z e. _V
11		1stexg 5814	. . . . . . . . 9 |- ( z e. _V -> ( 1st ` z ) e. _V )
12	10, 11	ax-mp 7	. . . . . . . 8 |- ( 1st ` z ) e. _V
13	9, 5, 12	fvmpt 5270	. . . . . . 7 |- ( z e. A -> ( ( x e. A |-> ( 1st ` x ) ) ` z ) = ( 1st ` z ) )
14	13	eqeq1d 2089	. . . . . 6 |- ( z e. A -> ( ( ( x e. A |-> ( 1st ` x ) ) ` z ) = y <-> ( 1st ` z ) = y ) )
15	14	rexbiia 2381	. . . . 5 |- ( E. z e. A ( ( x e. A |-> ( 1st ` x ) ) ` z ) = y <-> E. z e. A ( 1st ` z ) = y )
16	15	a1i 9	. . . 4 |- ( Rel A -> ( E. z e. A ( ( x e. A |-> ( 1st ` x ) ) ` z ) = y <-> E. z e. A ( 1st ` z ) = y ) )
17	8, 16	syl5rbb 191	. . 3 |- ( Rel A -> ( E. z e. A ( 1st ` z ) = y <-> y e. ran ( x e. A |-> ( 1st ` x ) ) ) )
18	1, 17	bitrd 186	. 2 |- ( Rel A -> ( y e. dom A <-> y e. ran ( x e. A |-> ( 1st ` x ) ) ) )
19	18	eqrdv 2079	1 |- ( Rel A -> dom A = ran ( x e. A |-> ( 1st ` x ) ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   e. wcel 1433   E.wrex 2349   _Vcvv 2601   |-> cmpt 3839   dom cdm 4363   ran crn 4364   Rel wrel 4368   Fn wfn 4917   ` cfv 4922   1stc1st 5785

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fo 4928  df-fv 4930  df-1st 5787  df-2nd 5788

This theorem is referenced by: (None)