Theorem reldm 6348

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 6347	. . 3 |- ( Rel A -> ( y e. dom A <-> E. z e. A ( 1st ` z ) = y ) )
2		vex 2805	. . . . . . 7 |- x e. _V
3		1stexg 6329	. . . . . . 7 |- ( x e. _V -> ( 1st ` x ) e. _V )
4	2, 3	ax-mp 5	. . . . . 6 |- ( 1st ` x ) e. _V
5		eqid 2231	. . . . . 6 |- ( x e. A |-> ( 1st ` x ) ) = ( x e. A |-> ( 1st ` x ) )
6	4, 5	fnmpti 5461	. . . . 5 |- ( x e. A |-> ( 1st ` x ) ) Fn A
7		fvelrnb 5693	. . . . 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 5639	. . . . . . . 8 |- ( x = z -> ( 1st ` x ) = ( 1st ` z ) )
10		vex 2805	. . . . . . . . 9 |- z e. _V
11		1stexg 6329	. . . . . . . . 9 |- ( z e. _V -> ( 1st ` z ) e. _V )
12	10, 11	ax-mp 5	. . . . . . . 8 |- ( 1st ` z ) e. _V
13	9, 5, 12	fvmpt 5723	. . . . . . 7 |- ( z e. A -> ( ( x e. A |-> ( 1st ` x ) ) ` z ) = ( 1st ` z ) )
14	13	eqeq1d 2240	. . . . . 6 |- ( z e. A -> ( ( ( x e. A |-> ( 1st ` x ) ) ` z ) = y <-> ( 1st ` z ) = y ) )
15	14	rexbiia 2547	. . . . 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 2229	1 |- ( Rel A -> dom A = ran ( x e. A |-> ( 1st ` x ) ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1397   e. wcel 2202   E.wrex 2511   _Vcvv 2802   |-> cmpt 4150   dom cdm 4725   ran crn 4726   Rel wrel 4730   Fn wfn 5321   ` cfv 5326   1stc1st 6300

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fo 5332  df-fv 5334  df-1st 6302  df-2nd 6303

This theorem is referenced by: (None)