Theorem basprssdmsets 17319

Description: The pair of the base index and another index is a subset of the domain of the structure obtained by replacing/adding a slot at the other index in a structure having a base slot. (Contributed by AV, 7-Jun-2021.) (Revised by AV, 16-Nov-2021.)

Hypotheses

Ref	Expression
basprssdmsets.s	⊢ (𝜑 → 𝑆 Struct 𝑋)
basprssdmsets.i	⊢ (𝜑 → 𝐼 ∈ 𝑈)
basprssdmsets.w	⊢ (𝜑 → 𝐸 ∈ 𝑊)
basprssdmsets.b	⊢ (𝜑 → (Base‘ndx) ∈ dom 𝑆)

Assertion

Ref	Expression
basprssdmsets	⊢ (𝜑 → {(Base‘ndx), 𝐼} ⊆ dom (𝑆 sSet ⟨𝐼, 𝐸⟩))

Proof of Theorem basprssdmsets

Step	Hyp	Ref	Expression
1		basprssdmsets.b	. . . . 5 ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝑆)
2	1	orcd 887	. . . 4 ⊢ (𝜑 → ((Base‘ndx) ∈ dom 𝑆 ∨ (Base‘ndx) ∈ {𝐼}))
3		elun 4103	. . . 4 ⊢ ((Base‘ndx) ∈ (dom 𝑆 ∪ {𝐼}) ↔ ((Base‘ndx) ∈ dom 𝑆 ∨ (Base‘ndx) ∈ {𝐼}))
4	2, 3	sylibr 237	. . 3 ⊢ (𝜑 → (Base‘ndx) ∈ (dom 𝑆 ∪ {𝐼}))
5		basprssdmsets.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑈)
6		snidg 4624	. . . . . 6 ⊢ (𝐼 ∈ 𝑈 → 𝐼 ∈ {𝐼})
7	5, 6	syl 18	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ {𝐼})
8	7	olcd 888	. . . 4 ⊢ (𝜑 → (𝐼 ∈ dom 𝑆 ∨ 𝐼 ∈ {𝐼}))
9		elun 4103	. . . 4 ⊢ (𝐼 ∈ (dom 𝑆 ∪ {𝐼}) ↔ (𝐼 ∈ dom 𝑆 ∨ 𝐼 ∈ {𝐼}))
10	8, 9	sylibr 237	. . 3 ⊢ (𝜑 → 𝐼 ∈ (dom 𝑆 ∪ {𝐼}))
11	4, 10	prssd 4786	. 2 ⊢ (𝜑 → {(Base‘ndx), 𝐼} ⊆ (dom 𝑆 ∪ {𝐼}))
12		basprssdmsets.s	. . . 4 ⊢ (𝜑 → 𝑆 Struct 𝑋)
13		structex 17248	. . . 4 ⊢ (𝑆 Struct 𝑋 → 𝑆 ∈ V)
14	12, 13	syl 18	. . 3 ⊢ (𝜑 → 𝑆 ∈ V)
15		basprssdmsets.w	. . 3 ⊢ (𝜑 → 𝐸 ∈ 𝑊)
16		setsdm 17268	. . 3 ⊢ ((𝑆 ∈ V ∧ 𝐸 ∈ 𝑊) → dom (𝑆 sSet ⟨𝐼, 𝐸⟩) = (dom 𝑆 ∪ {𝐼}))
17	14, 15, 16	syl2anc 596	. 2 ⊢ (𝜑 → dom (𝑆 sSet ⟨𝐼, 𝐸⟩) = (dom 𝑆 ∪ {𝐼}))
18	11, 17	sseqtrrd 3971	1 ⊢ (𝜑 → {(Base‘ndx), 𝐼} ⊆ dom (𝑆 sSet ⟨𝐼, 𝐸⟩))

This proof depends on syntax axioms:   → wi 4   ∨ wo 861   = wceq 1570   ∈ wcel 2145  Vcvv 3453   ∪ cun 3900   ⊆ wss 3902  {csn 4587  {cpr 4589  ⟨cop 4593   class class class wbr 5107  dom cdm 5659  ‘cfv 6537  (class class class)co 7417   Struct cstr 17244   sSet csts 17261  ndxcnx 17291  Basecbs 17307

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-res 5671  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-struct 17245  df-sets 17262

This theorem is used by:  setsvtx  29500  setsiedg  29501