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Theorem setsdm 15939
Description: The domain of a structure with replacement is the domain of the original structure extended by the index of the replacement. (Contributed by AV, 7-Jun-2021.)
Assertion
Ref Expression
setsdm ((𝐺𝑉𝐸𝑊) → dom (𝐺 sSet ⟨𝐼, 𝐸⟩) = (dom 𝐺 ∪ {𝐼}))

Proof of Theorem setsdm
StepHypRef Expression
1 opex 4962 . . . . 5 𝐼, 𝐸⟩ ∈ V
21a1i 11 . . . 4 (𝐸𝑊 → ⟨𝐼, 𝐸⟩ ∈ V)
3 setsvalg 15934 . . . 4 ((𝐺𝑉 ∧ ⟨𝐼, 𝐸⟩ ∈ V) → (𝐺 sSet ⟨𝐼, 𝐸⟩) = ((𝐺 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ {⟨𝐼, 𝐸⟩}))
42, 3sylan2 490 . . 3 ((𝐺𝑉𝐸𝑊) → (𝐺 sSet ⟨𝐼, 𝐸⟩) = ((𝐺 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ {⟨𝐼, 𝐸⟩}))
54dmeqd 5358 . 2 ((𝐺𝑉𝐸𝑊) → dom (𝐺 sSet ⟨𝐼, 𝐸⟩) = dom ((𝐺 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ {⟨𝐼, 𝐸⟩}))
6 dmun 5363 . . 3 dom ((𝐺 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ {⟨𝐼, 𝐸⟩}) = (dom (𝐺 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ dom {⟨𝐼, 𝐸⟩})
7 dmres 5454 . . . . 5 dom (𝐺 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) = ((V ∖ dom {⟨𝐼, 𝐸⟩}) ∩ dom 𝐺)
8 dmsnopg 5642 . . . . . . . . 9 (𝐸𝑊 → dom {⟨𝐼, 𝐸⟩} = {𝐼})
98adantl 481 . . . . . . . 8 ((𝐺𝑉𝐸𝑊) → dom {⟨𝐼, 𝐸⟩} = {𝐼})
109difeq2d 3761 . . . . . . 7 ((𝐺𝑉𝐸𝑊) → (V ∖ dom {⟨𝐼, 𝐸⟩}) = (V ∖ {𝐼}))
1110ineq1d 3846 . . . . . 6 ((𝐺𝑉𝐸𝑊) → ((V ∖ dom {⟨𝐼, 𝐸⟩}) ∩ dom 𝐺) = ((V ∖ {𝐼}) ∩ dom 𝐺))
12 incom 3838 . . . . . . 7 ((V ∖ {𝐼}) ∩ dom 𝐺) = (dom 𝐺 ∩ (V ∖ {𝐼}))
13 invdif 3901 . . . . . . 7 (dom 𝐺 ∩ (V ∖ {𝐼})) = (dom 𝐺 ∖ {𝐼})
1412, 13eqtri 2673 . . . . . 6 ((V ∖ {𝐼}) ∩ dom 𝐺) = (dom 𝐺 ∖ {𝐼})
1511, 14syl6eq 2701 . . . . 5 ((𝐺𝑉𝐸𝑊) → ((V ∖ dom {⟨𝐼, 𝐸⟩}) ∩ dom 𝐺) = (dom 𝐺 ∖ {𝐼}))
167, 15syl5eq 2697 . . . 4 ((𝐺𝑉𝐸𝑊) → dom (𝐺 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) = (dom 𝐺 ∖ {𝐼}))
1716, 9uneq12d 3801 . . 3 ((𝐺𝑉𝐸𝑊) → (dom (𝐺 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ dom {⟨𝐼, 𝐸⟩}) = ((dom 𝐺 ∖ {𝐼}) ∪ {𝐼}))
186, 17syl5eq 2697 . 2 ((𝐺𝑉𝐸𝑊) → dom ((𝐺 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ {⟨𝐼, 𝐸⟩}) = ((dom 𝐺 ∖ {𝐼}) ∪ {𝐼}))
19 undif1 4076 . . 3 ((dom 𝐺 ∖ {𝐼}) ∪ {𝐼}) = (dom 𝐺 ∪ {𝐼})
2019a1i 11 . 2 ((𝐺𝑉𝐸𝑊) → ((dom 𝐺 ∖ {𝐼}) ∪ {𝐼}) = (dom 𝐺 ∪ {𝐼}))
215, 18, 203eqtrd 2689 1 ((𝐺𝑉𝐸𝑊) → dom (𝐺 sSet ⟨𝐼, 𝐸⟩) = (dom 𝐺 ∪ {𝐼}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  cdif 3604  cun 3605  cin 3606  {csn 4210  cop 4216  dom cdm 5143  cres 5145  (class class class)co 6690   sSet csts 15902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-res 5155  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-sets 15911
This theorem is referenced by:  setsstruct2  15943  setsstructOLD  15946  basprssdmsets  15972
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