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Theorem setsdm 16165
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 5088 . . . . 5 𝐼, 𝐸⟩ ∈ V
21a1i 11 . . . 4 (𝐸𝑊 → ⟨𝐼, 𝐸⟩ ∈ V)
3 setsvalg 16160 . . . 4 ((𝐺𝑉 ∧ ⟨𝐼, 𝐸⟩ ∈ V) → (𝐺 sSet ⟨𝐼, 𝐸⟩) = ((𝐺 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ {⟨𝐼, 𝐸⟩}))
42, 3sylan2 586 . . 3 ((𝐺𝑉𝐸𝑊) → (𝐺 sSet ⟨𝐼, 𝐸⟩) = ((𝐺 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ {⟨𝐼, 𝐸⟩}))
54dmeqd 5494 . 2 ((𝐺𝑉𝐸𝑊) → dom (𝐺 sSet ⟨𝐼, 𝐸⟩) = dom ((𝐺 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ {⟨𝐼, 𝐸⟩}))
6 dmun 5499 . . 3 dom ((𝐺 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ {⟨𝐼, 𝐸⟩}) = (dom (𝐺 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ dom {⟨𝐼, 𝐸⟩})
7 dmres 5594 . . . . 5 dom (𝐺 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) = ((V ∖ dom {⟨𝐼, 𝐸⟩}) ∩ dom 𝐺)
8 dmsnopg 5790 . . . . . . . . 9 (𝐸𝑊 → dom {⟨𝐼, 𝐸⟩} = {𝐼})
98adantl 473 . . . . . . . 8 ((𝐺𝑉𝐸𝑊) → dom {⟨𝐼, 𝐸⟩} = {𝐼})
109difeq2d 3890 . . . . . . 7 ((𝐺𝑉𝐸𝑊) → (V ∖ dom {⟨𝐼, 𝐸⟩}) = (V ∖ {𝐼}))
1110ineq1d 3975 . . . . . 6 ((𝐺𝑉𝐸𝑊) → ((V ∖ dom {⟨𝐼, 𝐸⟩}) ∩ dom 𝐺) = ((V ∖ {𝐼}) ∩ dom 𝐺))
12 incom 3967 . . . . . . 7 ((V ∖ {𝐼}) ∩ dom 𝐺) = (dom 𝐺 ∩ (V ∖ {𝐼}))
13 invdif 4033 . . . . . . 7 (dom 𝐺 ∩ (V ∖ {𝐼})) = (dom 𝐺 ∖ {𝐼})
1412, 13eqtri 2787 . . . . . 6 ((V ∖ {𝐼}) ∩ dom 𝐺) = (dom 𝐺 ∖ {𝐼})
1511, 14syl6eq 2815 . . . . 5 ((𝐺𝑉𝐸𝑊) → ((V ∖ dom {⟨𝐼, 𝐸⟩}) ∩ dom 𝐺) = (dom 𝐺 ∖ {𝐼}))
167, 15syl5eq 2811 . . . 4 ((𝐺𝑉𝐸𝑊) → dom (𝐺 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) = (dom 𝐺 ∖ {𝐼}))
1716, 9uneq12d 3930 . . 3 ((𝐺𝑉𝐸𝑊) → (dom (𝐺 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ dom {⟨𝐼, 𝐸⟩}) = ((dom 𝐺 ∖ {𝐼}) ∪ {𝐼}))
186, 17syl5eq 2811 . 2 ((𝐺𝑉𝐸𝑊) → dom ((𝐺 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ {⟨𝐼, 𝐸⟩}) = ((dom 𝐺 ∖ {𝐼}) ∪ {𝐼}))
19 undif1 4203 . . 3 ((dom 𝐺 ∖ {𝐼}) ∪ {𝐼}) = (dom 𝐺 ∪ {𝐼})
2019a1i 11 . 2 ((𝐺𝑉𝐸𝑊) → ((dom 𝐺 ∖ {𝐼}) ∪ {𝐼}) = (dom 𝐺 ∪ {𝐼}))
215, 18, 203eqtrd 2803 1 ((𝐺𝑉𝐸𝑊) → dom (𝐺 sSet ⟨𝐼, 𝐸⟩) = (dom 𝐺 ∪ {𝐼}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  Vcvv 3350  cdif 3729  cun 3730  cin 3731  {csn 4334  cop 4340  dom cdm 5277  cres 5279  (class class class)co 6842   sSet csts 16128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-res 5289  df-iota 6031  df-fun 6070  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-sets 16137
This theorem is referenced by:  setsstruct2  16169  basprssdmsets  16197
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