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Theorem basprssdmsets 17259
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 (𝜑𝑆 Struct 𝑋)
basprssdmsets.i (𝜑𝐼𝑈)
basprssdmsets.w (𝜑𝐸𝑊)
basprssdmsets.b (𝜑 → (Base‘ndx) ∈ dom 𝑆)
Assertion
Ref Expression
basprssdmsets (𝜑 → {(Base‘ndx), 𝐼} ⊆ dom (𝑆 sSet ⟨𝐼, 𝐸⟩))

Proof of Theorem basprssdmsets
StepHypRef Expression
1 basprssdmsets.b . . . . 5 (𝜑 → (Base‘ndx) ∈ dom 𝑆)
21orcd 874 . . . 4 (𝜑 → ((Base‘ndx) ∈ dom 𝑆 ∨ (Base‘ndx) ∈ {𝐼}))
3 elun 4153 . . . 4 ((Base‘ndx) ∈ (dom 𝑆 ∪ {𝐼}) ↔ ((Base‘ndx) ∈ dom 𝑆 ∨ (Base‘ndx) ∈ {𝐼}))
42, 3sylibr 234 . . 3 (𝜑 → (Base‘ndx) ∈ (dom 𝑆 ∪ {𝐼}))
5 basprssdmsets.i . . . . . 6 (𝜑𝐼𝑈)
6 snidg 4660 . . . . . 6 (𝐼𝑈𝐼 ∈ {𝐼})
75, 6syl 17 . . . . 5 (𝜑𝐼 ∈ {𝐼})
87olcd 875 . . . 4 (𝜑 → (𝐼 ∈ dom 𝑆𝐼 ∈ {𝐼}))
9 elun 4153 . . . 4 (𝐼 ∈ (dom 𝑆 ∪ {𝐼}) ↔ (𝐼 ∈ dom 𝑆𝐼 ∈ {𝐼}))
108, 9sylibr 234 . . 3 (𝜑𝐼 ∈ (dom 𝑆 ∪ {𝐼}))
114, 10prssd 4822 . 2 (𝜑 → {(Base‘ndx), 𝐼} ⊆ (dom 𝑆 ∪ {𝐼}))
12 basprssdmsets.s . . . 4 (𝜑𝑆 Struct 𝑋)
13 structex 17187 . . . 4 (𝑆 Struct 𝑋𝑆 ∈ V)
1412, 13syl 17 . . 3 (𝜑𝑆 ∈ V)
15 basprssdmsets.w . . 3 (𝜑𝐸𝑊)
16 setsdm 17207 . . 3 ((𝑆 ∈ V ∧ 𝐸𝑊) → dom (𝑆 sSet ⟨𝐼, 𝐸⟩) = (dom 𝑆 ∪ {𝐼}))
1714, 15, 16syl2anc 584 . 2 (𝜑 → dom (𝑆 sSet ⟨𝐼, 𝐸⟩) = (dom 𝑆 ∪ {𝐼}))
1811, 17sseqtrrd 4021 1 (𝜑 → {(Base‘ndx), 𝐼} ⊆ dom (𝑆 sSet ⟨𝐼, 𝐸⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 848   = wceq 1540  wcel 2108  Vcvv 3480  cun 3949  wss 3951  {csn 4626  {cpr 4628  cop 4632   class class class wbr 5143  dom cdm 5685  cfv 6561  (class class class)co 7431   Struct cstr 17183   sSet csts 17200  ndxcnx 17230  Basecbs 17247
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-struct 17184  df-sets 17201
This theorem is referenced by:  setsvtx  29052  setsiedg  29053
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