Theorem basprssdmsets 16250

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 900	. . . 4 ⊢ (𝜑 → ((Base‘ndx) ∈ dom 𝑆 ∨ (Base‘ndx) ∈ {𝐼}))
3		elun 3951	. . . 4 ⊢ ((Base‘ndx) ∈ (dom 𝑆 ∪ {𝐼}) ↔ ((Base‘ndx) ∈ dom 𝑆 ∨ (Base‘ndx) ∈ {𝐼}))
4	2, 3	sylibr 226	. . 3 ⊢ (𝜑 → (Base‘ndx) ∈ (dom 𝑆 ∪ {𝐼}))
5		basprssdmsets.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑈)
6		snidg 4398	. . . . . 6 ⊢ (𝐼 ∈ 𝑈 → 𝐼 ∈ {𝐼})
7	5, 6	syl 17	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ {𝐼})
8	7	olcd 901	. . . 4 ⊢ (𝜑 → (𝐼 ∈ dom 𝑆 ∨ 𝐼 ∈ {𝐼}))
9		elun 3951	. . . 4 ⊢ (𝐼 ∈ (dom 𝑆 ∪ {𝐼}) ↔ (𝐼 ∈ dom 𝑆 ∨ 𝐼 ∈ {𝐼}))
10	8, 9	sylibr 226	. . 3 ⊢ (𝜑 → 𝐼 ∈ (dom 𝑆 ∪ {𝐼}))
11	4, 10	prssd 4541	. 2 ⊢ (𝜑 → {(Base‘ndx), 𝐼} ⊆ (dom 𝑆 ∪ {𝐼}))
12		basprssdmsets.s	. . . 4 ⊢ (𝜑 → 𝑆 Struct 𝑋)
13		structex 16195	. . . 4 ⊢ (𝑆 Struct 𝑋 → 𝑆 ∈ V)
14	12, 13	syl 17	. . 3 ⊢ (𝜑 → 𝑆 ∈ V)
15		basprssdmsets.w	. . 3 ⊢ (𝜑 → 𝐸 ∈ 𝑊)
16		setsdm 16218	. . 3 ⊢ ((𝑆 ∈ V ∧ 𝐸 ∈ 𝑊) → dom (𝑆 sSet ⟨𝐼, 𝐸⟩) = (dom 𝑆 ∪ {𝐼}))
17	14, 15, 16	syl2anc 580	. 2 ⊢ (𝜑 → dom (𝑆 sSet ⟨𝐼, 𝐸⟩) = (dom 𝑆 ∪ {𝐼}))
18	11, 17	sseqtr4d 3838	1 ⊢ (𝜑 → {(Base‘ndx), 𝐼} ⊆ dom (𝑆 sSet ⟨𝐼, 𝐸⟩))

Syntax hints:   → wi 4   ∨ wo 874   = wceq 1653   ∈ wcel 2157  Vcvv 3385   ∪ cun 3767   ⊆ wss 3769  {csn 4368  {cpr 4370  ⟨cop 4374   class class class wbr 4843  dom cdm 5312  ‘cfv 6101  (class class class)co 6878   Struct cstr 16180  ndxcnx 16181   sSet csts 16182  Basecbs 16184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097  ax-un 7183

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-res 5324  df-iota 6064  df-fun 6103  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-struct 16186  df-sets 16191

This theorem is referenced by:  setsvtx  26270  setsiedg  26271