Theorem reldm 6332

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 6331	. . 3 |- ( Rel A -> ( y e. dom A <-> E. z e. A ( 1st ` z ) = y ) )
2		vex 2802	. . . . . . 7 |- x e. _V
3		1stexg 6313	. . . . . . 7 |- ( x e. _V -> ( 1st ` x ) e. _V )
4	2, 3	ax-mp 5	. . . . . 6 |- ( 1st ` x ) e. _V
5		eqid 2229	. . . . . 6 |- ( x e. A |-> ( 1st ` x ) ) = ( x e. A |-> ( 1st ` x ) )
6	4, 5	fnmpti 5452	. . . . 5 |- ( x e. A |-> ( 1st ` x ) ) Fn A
7		fvelrnb 5681	. . . . 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 5	. . . 4 |- ( y e. ran ( x e. A |-> ( 1st ` x ) ) <-> E. z e. A ( ( x e. A |-> ( 1st ` x ) ) ` z ) = y )
9		fveq2 5627	. . . . . . . 8 |- ( x = z -> ( 1st ` x ) = ( 1st ` z ) )
10		vex 2802	. . . . . . . . 9 |- z e. _V
11		1stexg 6313	. . . . . . . . 9 |- ( z e. _V -> ( 1st ` z ) e. _V )
12	10, 11	ax-mp 5	. . . . . . . 8 |- ( 1st ` z ) e. _V
13	9, 5, 12	fvmpt 5711	. . . . . . 7 |- ( z e. A -> ( ( x e. A |-> ( 1st ` x ) ) ` z ) = ( 1st ` z ) )
14	13	eqeq1d 2238	. . . . . 6 |- ( z e. A -> ( ( ( x e. A |-> ( 1st ` x ) ) ` z ) = y <-> ( 1st ` z ) = y ) )
15	14	rexbiia 2545	. . . . 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	bitr2id 193	. . 3 |- ( Rel A -> ( E. z e. A ( 1st ` z ) = y <-> y e. ran ( x e. A |-> ( 1st ` x ) ) ) )
18	1, 17	bitrd 188	. 2 |- ( Rel A -> ( y e. dom A <-> y e. ran ( x e. A |-> ( 1st ` x ) ) ) )
19	18	eqrdv 2227	1 |- ( Rel A -> dom A = ran ( x e. A |-> ( 1st ` x ) ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   e. wcel 2200   E.wrex 2509   _Vcvv 2799   |-> cmpt 4145   dom cdm 4719   ran crn 4720   Rel wrel 4724   Fn wfn 5313   ` cfv 5318   1stc1st 6284

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fo 5324  df-fv 5326  df-1st 6286  df-2nd 6287

This theorem is referenced by: (None)