Theorem reldm 5970

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 5969	. . 3 |- ( Rel A -> ( y e. dom A <-> E. z e. A ( 1st ` z ) = y ) )
2		vex 2623	. . . . . . 7 |- x e. _V
3		1stexg 5952	. . . . . . 7 |- ( x e. _V -> ( 1st ` x ) e. _V )
4	2, 3	ax-mp 7	. . . . . 6 |- ( 1st ` x ) e. _V
5		eqid 2089	. . . . . 6 |- ( x e. A |-> ( 1st ` x ) ) = ( x e. A |-> ( 1st ` x ) )
6	4, 5	fnmpti 5155	. . . . 5 |- ( x e. A |-> ( 1st ` x ) ) Fn A
7		fvelrnb 5365	. . . . 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 5318	. . . . . . . 8 |- ( x = z -> ( 1st ` x ) = ( 1st ` z ) )
10		vex 2623	. . . . . . . . 9 |- z e. _V
11		1stexg 5952	. . . . . . . . 9 |- ( z e. _V -> ( 1st ` z ) e. _V )
12	10, 11	ax-mp 7	. . . . . . . 8 |- ( 1st ` z ) e. _V
13	9, 5, 12	fvmpt 5394	. . . . . . 7 |- ( z e. A -> ( ( x e. A |-> ( 1st ` x ) ) ` z ) = ( 1st ` z ) )
14	13	eqeq1d 2097	. . . . . 6 |- ( z e. A -> ( ( ( x e. A |-> ( 1st ` x ) ) ` z ) = y <-> ( 1st ` z ) = y ) )
15	14	rexbiia 2394	. . . . 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 192	. . 3 |- ( Rel A -> ( E. z e. A ( 1st ` z ) = y <-> y e. ran ( x e. A |-> ( 1st ` x ) ) ) )
18	1, 17	bitrd 187	. 2 |- ( Rel A -> ( y e. dom A <-> y e. ran ( x e. A |-> ( 1st ` x ) ) ) )
19	18	eqrdv 2087	1 |- ( Rel A -> dom A = ran ( x e. A |-> ( 1st ` x ) ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1290   e. wcel 1439   E.wrex 2361   _Vcvv 2620   |-> cmpt 3905   dom cdm 4452   ran crn 4453   Rel wrel 4457   Fn wfn 5023   ` cfv 5028   1stc1st 5923

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-fo 5034  df-fv 5036  df-1st 5925  df-2nd 5926

This theorem is referenced by: (None)