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Theorem basprssdmsets 16547
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 870 . . . 4 (𝜑 → ((Base‘ndx) ∈ dom 𝑆 ∨ (Base‘ndx) ∈ {𝐼}))
3 elun 4111 . . . 4 ((Base‘ndx) ∈ (dom 𝑆 ∪ {𝐼}) ↔ ((Base‘ndx) ∈ dom 𝑆 ∨ (Base‘ndx) ∈ {𝐼}))
42, 3sylibr 237 . . 3 (𝜑 → (Base‘ndx) ∈ (dom 𝑆 ∪ {𝐼}))
5 basprssdmsets.i . . . . . 6 (𝜑𝐼𝑈)
6 snidg 4584 . . . . . 6 (𝐼𝑈𝐼 ∈ {𝐼})
75, 6syl 17 . . . . 5 (𝜑𝐼 ∈ {𝐼})
87olcd 871 . . . 4 (𝜑 → (𝐼 ∈ dom 𝑆𝐼 ∈ {𝐼}))
9 elun 4111 . . . 4 (𝐼 ∈ (dom 𝑆 ∪ {𝐼}) ↔ (𝐼 ∈ dom 𝑆𝐼 ∈ {𝐼}))
108, 9sylibr 237 . . 3 (𝜑𝐼 ∈ (dom 𝑆 ∪ {𝐼}))
114, 10prssd 4740 . 2 (𝜑 → {(Base‘ndx), 𝐼} ⊆ (dom 𝑆 ∪ {𝐼}))
12 basprssdmsets.s . . . 4 (𝜑𝑆 Struct 𝑋)
13 structex 16492 . . . 4 (𝑆 Struct 𝑋𝑆 ∈ V)
1412, 13syl 17 . . 3 (𝜑𝑆 ∈ V)
15 basprssdmsets.w . . 3 (𝜑𝐸𝑊)
16 setsdm 16515 . . 3 ((𝑆 ∈ V ∧ 𝐸𝑊) → dom (𝑆 sSet ⟨𝐼, 𝐸⟩) = (dom 𝑆 ∪ {𝐼}))
1714, 15, 16syl2anc 587 . 2 (𝜑 → dom (𝑆 sSet ⟨𝐼, 𝐸⟩) = (dom 𝑆 ∪ {𝐼}))
1811, 17sseqtrrd 3994 1 (𝜑 → {(Base‘ndx), 𝐼} ⊆ dom (𝑆 sSet ⟨𝐼, 𝐸⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 844   = wceq 1538  wcel 2115  Vcvv 3480  cun 3917  wss 3919  {csn 4550  {cpr 4552  cop 4556   class class class wbr 5053  dom cdm 5543  cfv 6344  (class class class)co 7146   Struct cstr 16477  ndxcnx 16478   sSet csts 16479  Basecbs 16481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318  ax-un 7452
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-res 5555  df-iota 6303  df-fun 6346  df-fv 6352  df-ov 7149  df-oprab 7150  df-mpo 7151  df-struct 16483  df-sets 16488
This theorem is referenced by:  setsvtx  26826  setsiedg  26827
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