Theorem shftdm 11507

Description: Domain of a relation shifted by A. The set on the right is more commonly notated as ( dom F + A ) (meaning add A to every element of dom F). (Contributed by Mario Carneiro, 3-Nov-2013.)

Hypothesis

Ref	Expression
shftfval.1	|- F e. _V

Assertion

Ref	Expression
shftdm	|- ( A e. CC -> dom ( F shift A ) = { x e. CC | ( x - A ) e. dom F } )

Distinct variable groups:   x, A   x, F

Proof of Theorem shftdm

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		shftfval.1	. . . 4 |- F e. _V
2	1	shftfval 11506	. . 3 |- ( A e. CC -> ( F shift A ) = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )
3	2	dmeqd 4958	. 2 |- ( A e. CC -> dom ( F shift A ) = dom { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )
4		19.42v 1956	. . . . 5 |- ( E. y ( x e. CC /\ ( x - A ) F y ) <-> ( x e. CC /\ E. y ( x - A ) F y ) )
5		simpr 110	. . . . . . . 8 |- ( ( A e. CC /\ x e. CC ) -> x e. CC )
6		simpl 109	. . . . . . . 8 |- ( ( A e. CC /\ x e. CC ) -> A e. CC )
7	5, 6	subcld 8584	. . . . . . 7 |- ( ( A e. CC /\ x e. CC ) -> ( x - A ) e. CC )
8		eldmg 4951	. . . . . . 7 |- ( ( x - A ) e. CC -> ( ( x - A ) e. dom F <-> E. y ( x - A ) F y ) )
9	7, 8	syl 14	. . . . . 6 |- ( ( A e. CC /\ x e. CC ) -> ( ( x - A ) e. dom F <-> E. y ( x - A ) F y ) )
10	9	pm5.32da 452	. . . . 5 |- ( A e. CC -> ( ( x e. CC /\ ( x - A ) e. dom F ) <-> ( x e. CC /\ E. y ( x - A ) F y ) ) )
11	4, 10	bitr4id 199	. . . 4 |- ( A e. CC -> ( E. y ( x e. CC /\ ( x - A ) F y ) <-> ( x e. CC /\ ( x - A ) e. dom F ) ) )
12	11	abbidv 2352	. . 3 |- ( A e. CC -> { x | E. y ( x e. CC /\ ( x - A ) F y ) } = { x | ( x e. CC /\ ( x - A ) e. dom F ) } )
13		dmopab 4967	. . 3 |- dom { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } = { x | E. y ( x e. CC /\ ( x - A ) F y ) }
14		df-rab 2529	. . 3 |- { x e. CC | ( x - A ) e. dom F } = { x | ( x e. CC /\ ( x - A ) e. dom F ) }
15	12, 13, 14	3eqtr4g 2290	. 2 |- ( A e. CC -> dom { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } = { x e. CC | ( x - A ) e. dom F } )
16	3, 15	eqtrd 2265	1 |- ( A e. CC -> dom ( F shift A ) = { x e. CC | ( x - A ) e. dom F } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2203   {cab 2218   {crab 2524   _Vcvv 2813   class class class wbr 4109   {copab 4170   dom cdm 4749  (class class class)co 6050   CCcc 8125   - cmin 8444   shift cshi 11499

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-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-resscn 8219  ax-1cn 8220  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-sub 8446  df-shft 11500

This theorem is referenced by:  shftfn  11509