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Theorem basprssdmsets 15906
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 407 . . . 4 (𝜑 → ((Base‘ndx) ∈ dom 𝑆 ∨ (Base‘ndx) ∈ {𝐼}))
3 elun 3745 . . . 4 ((Base‘ndx) ∈ (dom 𝑆 ∪ {𝐼}) ↔ ((Base‘ndx) ∈ dom 𝑆 ∨ (Base‘ndx) ∈ {𝐼}))
42, 3sylibr 224 . . 3 (𝜑 → (Base‘ndx) ∈ (dom 𝑆 ∪ {𝐼}))
5 basprssdmsets.i . . . . . 6 (𝜑𝐼𝑈)
6 snidg 4197 . . . . . 6 (𝐼𝑈𝐼 ∈ {𝐼})
75, 6syl 17 . . . . 5 (𝜑𝐼 ∈ {𝐼})
87olcd 408 . . . 4 (𝜑 → (𝐼 ∈ dom 𝑆𝐼 ∈ {𝐼}))
9 elun 3745 . . . 4 (𝐼 ∈ (dom 𝑆 ∪ {𝐼}) ↔ (𝐼 ∈ dom 𝑆𝐼 ∈ {𝐼}))
108, 9sylibr 224 . . 3 (𝜑𝐼 ∈ (dom 𝑆 ∪ {𝐼}))
114, 10prssd 4345 . 2 (𝜑 → {(Base‘ndx), 𝐼} ⊆ (dom 𝑆 ∪ {𝐼}))
12 basprssdmsets.s . . . 4 (𝜑𝑆 Struct 𝑋)
13 structex 15849 . . . 4 (𝑆 Struct 𝑋𝑆 ∈ V)
1412, 13syl 17 . . 3 (𝜑𝑆 ∈ V)
15 basprssdmsets.w . . 3 (𝜑𝐸𝑊)
16 setsdm 15873 . . 3 ((𝑆 ∈ V ∧ 𝐸𝑊) → dom (𝑆 sSet ⟨𝐼, 𝐸⟩) = (dom 𝑆 ∪ {𝐼}))
1714, 15, 16syl2anc 692 . 2 (𝜑 → dom (𝑆 sSet ⟨𝐼, 𝐸⟩) = (dom 𝑆 ∪ {𝐼}))
1811, 17sseqtr4d 3634 1 (𝜑 → {(Base‘ndx), 𝐼} ⊆ dom (𝑆 sSet ⟨𝐼, 𝐸⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383   = wceq 1481  wcel 1988  Vcvv 3195  cun 3565  wss 3567  {csn 4168  {cpr 4170  cop 4174   class class class wbr 4644  dom cdm 5104  cfv 5876  (class class class)co 6635   Struct cstr 15834  ndxcnx 15835   sSet csts 15836  Basecbs 15838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-res 5116  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-struct 15840  df-sets 15845
This theorem is referenced by:  setsvtx  25908  setsiedg  25909
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