Theorem basprssdmsets 17134

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 873	. . . 4 ⊢ (𝜑 → ((Base‘ndx) ∈ dom 𝑆 ∨ (Base‘ndx) ∈ {𝐼}))
3		elun 4102	. . . 4 ⊢ ((Base‘ndx) ∈ (dom 𝑆 ∪ {𝐼}) ↔ ((Base‘ndx) ∈ dom 𝑆 ∨ (Base‘ndx) ∈ {𝐼}))
4	2, 3	sylibr 234	. . 3 ⊢ (𝜑 → (Base‘ndx) ∈ (dom 𝑆 ∪ {𝐼}))
5		basprssdmsets.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑈)
6		snidg 4612	. . . . . 6 ⊢ (𝐼 ∈ 𝑈 → 𝐼 ∈ {𝐼})
7	5, 6	syl 17	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ {𝐼})
8	7	olcd 874	. . . 4 ⊢ (𝜑 → (𝐼 ∈ dom 𝑆 ∨ 𝐼 ∈ {𝐼}))
9		elun 4102	. . . 4 ⊢ (𝐼 ∈ (dom 𝑆 ∪ {𝐼}) ↔ (𝐼 ∈ dom 𝑆 ∨ 𝐼 ∈ {𝐼}))
10	8, 9	sylibr 234	. . 3 ⊢ (𝜑 → 𝐼 ∈ (dom 𝑆 ∪ {𝐼}))
11	4, 10	prssd 4773	. 2 ⊢ (𝜑 → {(Base‘ndx), 𝐼} ⊆ (dom 𝑆 ∪ {𝐼}))
12		basprssdmsets.s	. . . 4 ⊢ (𝜑 → 𝑆 Struct 𝑋)
13		structex 17063	. . . 4 ⊢ (𝑆 Struct 𝑋 → 𝑆 ∈ V)
14	12, 13	syl 17	. . 3 ⊢ (𝜑 → 𝑆 ∈ V)
15		basprssdmsets.w	. . 3 ⊢ (𝜑 → 𝐸 ∈ 𝑊)
16		setsdm 17083	. . 3 ⊢ ((𝑆 ∈ V ∧ 𝐸 ∈ 𝑊) → dom (𝑆 sSet ⟨𝐼, 𝐸⟩) = (dom 𝑆 ∪ {𝐼}))
17	14, 15, 16	syl2anc 584	. 2 ⊢ (𝜑 → dom (𝑆 sSet ⟨𝐼, 𝐸⟩) = (dom 𝑆 ∪ {𝐼}))
18	11, 17	sseqtrrd 3968	1 ⊢ (𝜑 → {(Base‘ndx), 𝐼} ⊆ dom (𝑆 sSet ⟨𝐼, 𝐸⟩))

Syntax hints:   → wi 4   ∨ wo 847   = wceq 1541   ∈ wcel 2113  Vcvv 3437   ∪ cun 3896   ⊆ wss 3898  {csn 4575  {cpr 4577  ⟨cop 4581   class class class wbr 5093  dom cdm 5619  ‘cfv 6486  (class class class)co 7352   Struct cstr 17059   sSet csts 17076  ndxcnx 17106  Basecbs 17122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-res 5631  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-struct 17060  df-sets 17077

This theorem is referenced by:  setsvtx  29015  setsiedg  29016