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Theorem setsdm 15724
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 4859 . . . . 5 𝐼, 𝐸⟩ ∈ V
21a1i 11 . . . 4 (𝐸𝑊 → ⟨𝐼, 𝐸⟩ ∈ V)
3 setsvalg 15719 . . . 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 5248 . 2 ((𝐺𝑉𝐸𝑊) → dom (𝐺 sSet ⟨𝐼, 𝐸⟩) = dom ((𝐺 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ {⟨𝐼, 𝐸⟩}))
6 dmun 5253 . . 3 dom ((𝐺 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ {⟨𝐼, 𝐸⟩}) = (dom (𝐺 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ dom {⟨𝐼, 𝐸⟩})
7 dmres 5339 . . . . 5 dom (𝐺 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) = ((V ∖ dom {⟨𝐼, 𝐸⟩}) ∩ dom 𝐺)
8 dmsnopg 5524 . . . . . . . . 9 (𝐸𝑊 → dom {⟨𝐼, 𝐸⟩} = {𝐼})
98adantl 481 . . . . . . . 8 ((𝐺𝑉𝐸𝑊) → dom {⟨𝐼, 𝐸⟩} = {𝐼})
109difeq2d 3690 . . . . . . 7 ((𝐺𝑉𝐸𝑊) → (V ∖ dom {⟨𝐼, 𝐸⟩}) = (V ∖ {𝐼}))
1110ineq1d 3775 . . . . . 6 ((𝐺𝑉𝐸𝑊) → ((V ∖ dom {⟨𝐼, 𝐸⟩}) ∩ dom 𝐺) = ((V ∖ {𝐼}) ∩ dom 𝐺))
12 incom 3767 . . . . . . 7 ((V ∖ {𝐼}) ∩ dom 𝐺) = (dom 𝐺 ∩ (V ∖ {𝐼}))
13 invdif 3827 . . . . . . 7 (dom 𝐺 ∩ (V ∖ {𝐼})) = (dom 𝐺 ∖ {𝐼})
1412, 13eqtri 2632 . . . . . 6 ((V ∖ {𝐼}) ∩ dom 𝐺) = (dom 𝐺 ∖ {𝐼})
1511, 14syl6eq 2660 . . . . 5 ((𝐺𝑉𝐸𝑊) → ((V ∖ dom {⟨𝐼, 𝐸⟩}) ∩ dom 𝐺) = (dom 𝐺 ∖ {𝐼}))
167, 15syl5eq 2656 . . . 4 ((𝐺𝑉𝐸𝑊) → dom (𝐺 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) = (dom 𝐺 ∖ {𝐼}))
1716, 9uneq12d 3730 . . 3 ((𝐺𝑉𝐸𝑊) → (dom (𝐺 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ dom {⟨𝐼, 𝐸⟩}) = ((dom 𝐺 ∖ {𝐼}) ∪ {𝐼}))
186, 17syl5eq 2656 . 2 ((𝐺𝑉𝐸𝑊) → dom ((𝐺 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ {⟨𝐼, 𝐸⟩}) = ((dom 𝐺 ∖ {𝐼}) ∪ {𝐼}))
19 undif1 3995 . . 3 ((dom 𝐺 ∖ {𝐼}) ∪ {𝐼}) = (dom 𝐺 ∪ {𝐼})
2019a1i 11 . 2 ((𝐺𝑉𝐸𝑊) → ((dom 𝐺 ∖ {𝐼}) ∪ {𝐼}) = (dom 𝐺 ∪ {𝐼}))
215, 18, 203eqtrd 2648 1 ((𝐺𝑉𝐸𝑊) → dom (𝐺 sSet ⟨𝐼, 𝐸⟩) = (dom 𝐺 ∪ {𝐼}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  Vcvv 3173  cdif 3537  cun 3538  cin 3539  {csn 4125  cop 4131  dom cdm 5038  cres 5040  (class class class)co 6549   sSet csts 15693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-res 5050  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-sets 15701
This theorem is referenced by:  setsstruct  15727  uhgrstrrepelem  25744
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