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Theorem shftdm 10221
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 ∈ V
Assertion
Ref Expression
shftdm (𝐴 ∈ ℂ → dom (𝐹 shift 𝐴) = {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ dom 𝐹})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem shftdm
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 shftfval.1 . . . 4 𝐹 ∈ V
21shftfval 10220 . . 3 (𝐴 ∈ ℂ → (𝐹 shift 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})
32dmeqd 4626 . 2 (𝐴 ∈ ℂ → dom (𝐹 shift 𝐴) = dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})
4 simpr 108 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
5 simpl 107 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ ℂ)
64, 5subcld 7772 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑥𝐴) ∈ ℂ)
7 eldmg 4619 . . . . . . 7 ((𝑥𝐴) ∈ ℂ → ((𝑥𝐴) ∈ dom 𝐹 ↔ ∃𝑦(𝑥𝐴)𝐹𝑦))
86, 7syl 14 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝑥𝐴) ∈ dom 𝐹 ↔ ∃𝑦(𝑥𝐴)𝐹𝑦))
98pm5.32da 440 . . . . 5 (𝐴 ∈ ℂ → ((𝑥 ∈ ℂ ∧ (𝑥𝐴) ∈ dom 𝐹) ↔ (𝑥 ∈ ℂ ∧ ∃𝑦(𝑥𝐴)𝐹𝑦)))
10 19.42v 1834 . . . . 5 (∃𝑦(𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦) ↔ (𝑥 ∈ ℂ ∧ ∃𝑦(𝑥𝐴)𝐹𝑦))
119, 10syl6rbbr 197 . . . 4 (𝐴 ∈ ℂ → (∃𝑦(𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦) ↔ (𝑥 ∈ ℂ ∧ (𝑥𝐴) ∈ dom 𝐹)))
1211abbidv 2205 . . 3 (𝐴 ∈ ℂ → {𝑥 ∣ ∃𝑦(𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)} = {𝑥 ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴) ∈ dom 𝐹)})
13 dmopab 4635 . . 3 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)} = {𝑥 ∣ ∃𝑦(𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)}
14 df-rab 2368 . . 3 {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ dom 𝐹} = {𝑥 ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴) ∈ dom 𝐹)}
1512, 13, 143eqtr4g 2145 . 2 (𝐴 ∈ ℂ → dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)} = {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ dom 𝐹})
163, 15eqtrd 2120 1 (𝐴 ∈ ℂ → dom (𝐹 shift 𝐴) = {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ dom 𝐹})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wex 1426  wcel 1438  {cab 2074  {crab 2363  Vcvv 2619   class class class wbr 3837  {copab 3890  dom cdm 4428  (class class class)co 5634  cc 7327  cmin 7632   shift cshi 10213
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-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-resscn 7416  ax-1cn 7417  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-distr 7428  ax-i2m1 7429  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-sub 7634  df-shft 10214
This theorem is referenced by:  shftfn  10223
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