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Theorem basprssdmsets 17159
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 (πœ‘ β†’ 𝑆 Struct 𝑋)
basprssdmsets.i (πœ‘ β†’ 𝐼 ∈ π‘ˆ)
basprssdmsets.w (πœ‘ β†’ 𝐸 ∈ π‘Š)
basprssdmsets.b (πœ‘ β†’ (Baseβ€˜ndx) ∈ dom 𝑆)
Assertion
Ref Expression
basprssdmsets (πœ‘ β†’ {(Baseβ€˜ndx), 𝐼} βŠ† dom (𝑆 sSet ⟨𝐼, 𝐸⟩))

Proof of Theorem basprssdmsets
StepHypRef Expression
1 basprssdmsets.b . . . . 5 (πœ‘ β†’ (Baseβ€˜ndx) ∈ dom 𝑆)
21orcd 871 . . . 4 (πœ‘ β†’ ((Baseβ€˜ndx) ∈ dom 𝑆 ∨ (Baseβ€˜ndx) ∈ {𝐼}))
3 elun 4148 . . . 4 ((Baseβ€˜ndx) ∈ (dom 𝑆 βˆͺ {𝐼}) ↔ ((Baseβ€˜ndx) ∈ dom 𝑆 ∨ (Baseβ€˜ndx) ∈ {𝐼}))
42, 3sylibr 233 . . 3 (πœ‘ β†’ (Baseβ€˜ndx) ∈ (dom 𝑆 βˆͺ {𝐼}))
5 basprssdmsets.i . . . . . 6 (πœ‘ β†’ 𝐼 ∈ π‘ˆ)
6 snidg 4662 . . . . . 6 (𝐼 ∈ π‘ˆ β†’ 𝐼 ∈ {𝐼})
75, 6syl 17 . . . . 5 (πœ‘ β†’ 𝐼 ∈ {𝐼})
87olcd 872 . . . 4 (πœ‘ β†’ (𝐼 ∈ dom 𝑆 ∨ 𝐼 ∈ {𝐼}))
9 elun 4148 . . . 4 (𝐼 ∈ (dom 𝑆 βˆͺ {𝐼}) ↔ (𝐼 ∈ dom 𝑆 ∨ 𝐼 ∈ {𝐼}))
108, 9sylibr 233 . . 3 (πœ‘ β†’ 𝐼 ∈ (dom 𝑆 βˆͺ {𝐼}))
114, 10prssd 4825 . 2 (πœ‘ β†’ {(Baseβ€˜ndx), 𝐼} βŠ† (dom 𝑆 βˆͺ {𝐼}))
12 basprssdmsets.s . . . 4 (πœ‘ β†’ 𝑆 Struct 𝑋)
13 structex 17085 . . . 4 (𝑆 Struct 𝑋 β†’ 𝑆 ∈ V)
1412, 13syl 17 . . 3 (πœ‘ β†’ 𝑆 ∈ V)
15 basprssdmsets.w . . 3 (πœ‘ β†’ 𝐸 ∈ π‘Š)
16 setsdm 17105 . . 3 ((𝑆 ∈ V ∧ 𝐸 ∈ π‘Š) β†’ dom (𝑆 sSet ⟨𝐼, 𝐸⟩) = (dom 𝑆 βˆͺ {𝐼}))
1714, 15, 16syl2anc 584 . 2 (πœ‘ β†’ dom (𝑆 sSet ⟨𝐼, 𝐸⟩) = (dom 𝑆 βˆͺ {𝐼}))
1811, 17sseqtrrd 4023 1 (πœ‘ β†’ {(Baseβ€˜ndx), 𝐼} βŠ† dom (𝑆 sSet ⟨𝐼, 𝐸⟩))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∨ wo 845   = wceq 1541   ∈ wcel 2106  Vcvv 3474   βˆͺ cun 3946   βŠ† wss 3948  {csn 4628  {cpr 4630  βŸ¨cop 4634   class class class wbr 5148  dom cdm 5676  β€˜cfv 6543  (class class class)co 7411   Struct cstr 17081   sSet csts 17098  ndxcnx 17128  Basecbs 17146
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-res 5688  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-struct 17082  df-sets 17099
This theorem is referenced by:  setsvtx  28333  setsiedg  28334
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