Theorem basprssdmsets 16552

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 869	. . . 4 ⊢ (𝜑 → ((Base‘ndx) ∈ dom 𝑆 ∨ (Base‘ndx) ∈ {𝐼}))
3		elun 4128	. . . 4 ⊢ ((Base‘ndx) ∈ (dom 𝑆 ∪ {𝐼}) ↔ ((Base‘ndx) ∈ dom 𝑆 ∨ (Base‘ndx) ∈ {𝐼}))
4	2, 3	sylibr 236	. . 3 ⊢ (𝜑 → (Base‘ndx) ∈ (dom 𝑆 ∪ {𝐼}))
5		basprssdmsets.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑈)
6		snidg 4602	. . . . . 6 ⊢ (𝐼 ∈ 𝑈 → 𝐼 ∈ {𝐼})
7	5, 6	syl 17	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ {𝐼})
8	7	olcd 870	. . . 4 ⊢ (𝜑 → (𝐼 ∈ dom 𝑆 ∨ 𝐼 ∈ {𝐼}))
9		elun 4128	. . . 4 ⊢ (𝐼 ∈ (dom 𝑆 ∪ {𝐼}) ↔ (𝐼 ∈ dom 𝑆 ∨ 𝐼 ∈ {𝐼}))
10	8, 9	sylibr 236	. . 3 ⊢ (𝜑 → 𝐼 ∈ (dom 𝑆 ∪ {𝐼}))
11	4, 10	prssd 4758	. 2 ⊢ (𝜑 → {(Base‘ndx), 𝐼} ⊆ (dom 𝑆 ∪ {𝐼}))
12		basprssdmsets.s	. . . 4 ⊢ (𝜑 → 𝑆 Struct 𝑋)
13		structex 16497	. . . 4 ⊢ (𝑆 Struct 𝑋 → 𝑆 ∈ V)
14	12, 13	syl 17	. . 3 ⊢ (𝜑 → 𝑆 ∈ V)
15		basprssdmsets.w	. . 3 ⊢ (𝜑 → 𝐸 ∈ 𝑊)
16		setsdm 16520	. . 3 ⊢ ((𝑆 ∈ V ∧ 𝐸 ∈ 𝑊) → dom (𝑆 sSet ⟨𝐼, 𝐸⟩) = (dom 𝑆 ∪ {𝐼}))
17	14, 15, 16	syl2anc 586	. 2 ⊢ (𝜑 → dom (𝑆 sSet ⟨𝐼, 𝐸⟩) = (dom 𝑆 ∪ {𝐼}))
18	11, 17	sseqtrrd 4011	1 ⊢ (𝜑 → {(Base‘ndx), 𝐼} ⊆ dom (𝑆 sSet ⟨𝐼, 𝐸⟩))

Syntax hints:   → wi 4   ∨ wo 843   = wceq 1536   ∈ wcel 2113  Vcvv 3497   ∪ cun 3937   ⊆ wss 3939  {csn 4570  {cpr 4572  ⟨cop 4576   class class class wbr 5069  dom cdm 5558  ‘cfv 6358  (class class class)co 7159   Struct cstr 16482  ndxcnx 16483   sSet csts 16484  Basecbs 16486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-res 5570  df-iota 6317  df-fun 6360  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-struct 16488  df-sets 16493

This theorem is referenced by:  setsvtx  26823  setsiedg  26824