Theorem setsdm 15892

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	⊢ ((𝐺 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → dom (𝐺 sSet ⟨𝐼, 𝐸⟩) = (dom 𝐺 ∪ {𝐼}))

Proof of Theorem setsdm

Step	Hyp	Ref	Expression
1		opex 4932	. . . . 5 ⊢ ⟨𝐼, 𝐸⟩ ∈ V
2	1	a1i 11	. . . 4 ⊢ (𝐸 ∈ 𝑊 → ⟨𝐼, 𝐸⟩ ∈ V)
3		setsvalg 15887	. . . 4 ⊢ ((𝐺 ∈ 𝑉 ∧ ⟨𝐼, 𝐸⟩ ∈ V) → (𝐺 sSet ⟨𝐼, 𝐸⟩) = ((𝐺 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ {⟨𝐼, 𝐸⟩}))
4	2, 3	sylan2 491	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝐺 sSet ⟨𝐼, 𝐸⟩) = ((𝐺 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ {⟨𝐼, 𝐸⟩}))
5	4	dmeqd 5326	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → dom (𝐺 sSet ⟨𝐼, 𝐸⟩) = dom ((𝐺 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ {⟨𝐼, 𝐸⟩}))
6		dmun 5331	. . 3 ⊢ dom ((𝐺 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ {⟨𝐼, 𝐸⟩}) = (dom (𝐺 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ dom {⟨𝐼, 𝐸⟩})
7		dmres 5419	. . . . 5 ⊢ dom (𝐺 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) = ((V ∖ dom {⟨𝐼, 𝐸⟩}) ∩ dom 𝐺)
8		dmsnopg 5606	. . . . . . . . 9 ⊢ (𝐸 ∈ 𝑊 → dom {⟨𝐼, 𝐸⟩} = {𝐼})
9	8	adantl 482	. . . . . . . 8 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → dom {⟨𝐼, 𝐸⟩} = {𝐼})
10	9	difeq2d 3728	. . . . . . 7 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (V ∖ dom {⟨𝐼, 𝐸⟩}) = (V ∖ {𝐼}))
11	10	ineq1d 3813	. . . . . 6 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ((V ∖ dom {⟨𝐼, 𝐸⟩}) ∩ dom 𝐺) = ((V ∖ {𝐼}) ∩ dom 𝐺))
12		incom 3805	. . . . . . 7 ⊢ ((V ∖ {𝐼}) ∩ dom 𝐺) = (dom 𝐺 ∩ (V ∖ {𝐼}))
13		invdif 3868	. . . . . . 7 ⊢ (dom 𝐺 ∩ (V ∖ {𝐼})) = (dom 𝐺 ∖ {𝐼})
14	12, 13	eqtri 2644	. . . . . 6 ⊢ ((V ∖ {𝐼}) ∩ dom 𝐺) = (dom 𝐺 ∖ {𝐼})
15	11, 14	syl6eq 2672	. . . . 5 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ((V ∖ dom {⟨𝐼, 𝐸⟩}) ∩ dom 𝐺) = (dom 𝐺 ∖ {𝐼}))
16	7, 15	syl5eq 2668	. . . 4 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → dom (𝐺 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) = (dom 𝐺 ∖ {𝐼}))
17	16, 9	uneq12d 3768	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (dom (𝐺 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ dom {⟨𝐼, 𝐸⟩}) = ((dom 𝐺 ∖ {𝐼}) ∪ {𝐼}))
18	6, 17	syl5eq 2668	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → dom ((𝐺 ↾ (V ∖ dom {⟨𝐼, 𝐸⟩})) ∪ {⟨𝐼, 𝐸⟩}) = ((dom 𝐺 ∖ {𝐼}) ∪ {𝐼}))
19		undif1 4043	. . 3 ⊢ ((dom 𝐺 ∖ {𝐼}) ∪ {𝐼}) = (dom 𝐺 ∪ {𝐼})
20	19	a1i 11	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ((dom 𝐺 ∖ {𝐼}) ∪ {𝐼}) = (dom 𝐺 ∪ {𝐼}))
21	5, 18, 20	3eqtrd 2660	1 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → dom (𝐺 sSet ⟨𝐼, 𝐸⟩) = (dom 𝐺 ∪ {𝐼}))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ∩ cin 3573  {csn 4177  ⟨cop 4183  dom cdm 5114   ↾ cres 5116  (class class class)co 6650   sSet csts 15855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-sets 15864

This theorem is referenced by:  setsstruct2  15896  setsstructOLD  15899  basprssdmsets  15925