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Theorem shftdm 9423
 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 9422 . . 3 (𝐴 ∈ ℂ → (𝐹 shift 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})
32dmeqd 4537 . 2 (𝐴 ∈ ℂ → dom (𝐹 shift 𝐴) = dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})
4 simpr 103 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
5 simpl 102 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ ℂ)
64, 5subcld 7322 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑥𝐴) ∈ ℂ)
7 eldmg 4530 . . . . . . 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 425 . . . . 5 (𝐴 ∈ ℂ → ((𝑥 ∈ ℂ ∧ (𝑥𝐴) ∈ dom 𝐹) ↔ (𝑥 ∈ ℂ ∧ ∃𝑦(𝑥𝐴)𝐹𝑦)))
10 19.42v 1786 . . . . 5 (∃𝑦(𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦) ↔ (𝑥 ∈ ℂ ∧ ∃𝑦(𝑥𝐴)𝐹𝑦))
119, 10syl6rbbr 188 . . . 4 (𝐴 ∈ ℂ → (∃𝑦(𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦) ↔ (𝑥 ∈ ℂ ∧ (𝑥𝐴) ∈ dom 𝐹)))
1211abbidv 2155 . . 3 (𝐴 ∈ ℂ → {𝑥 ∣ ∃𝑦(𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)} = {𝑥 ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴) ∈ dom 𝐹)})
13 dmopab 4546 . . 3 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)} = {𝑥 ∣ ∃𝑦(𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)}
14 df-rab 2315 . . 3 {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ dom 𝐹} = {𝑥 ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴) ∈ dom 𝐹)}
1512, 13, 143eqtr4g 2097 . 2 (𝐴 ∈ ℂ → dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)} = {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ dom 𝐹})
163, 15eqtrd 2072 1 (𝐴 ∈ ℂ → dom (𝐹 shift 𝐴) = {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ dom 𝐹})
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243  ∃wex 1381   ∈ wcel 1393  {cab 2026  {crab 2310  Vcvv 2557   class class class wbr 3764  {copab 3817  dom cdm 4345  (class class class)co 5512  ℂcc 6887   − cmin 7182   shift cshi 9415 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-resscn 6976  ax-1cn 6977  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-sub 7184  df-shft 9416 This theorem is referenced by:  shftfn  9425
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