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Theorem basprssdmsets 17162
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 870 . . . 4 (πœ‘ β†’ ((Baseβ€˜ndx) ∈ dom 𝑆 ∨ (Baseβ€˜ndx) ∈ {𝐼}))
3 elun 4149 . . . 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 4663 . . . . . 6 (𝐼 ∈ π‘ˆ β†’ 𝐼 ∈ {𝐼})
75, 6syl 17 . . . . 5 (πœ‘ β†’ 𝐼 ∈ {𝐼})
87olcd 871 . . . 4 (πœ‘ β†’ (𝐼 ∈ dom 𝑆 ∨ 𝐼 ∈ {𝐼}))
9 elun 4149 . . . 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 4826 . 2 (πœ‘ β†’ {(Baseβ€˜ndx), 𝐼} βŠ† (dom 𝑆 βˆͺ {𝐼}))
12 basprssdmsets.s . . . 4 (πœ‘ β†’ 𝑆 Struct 𝑋)
13 structex 17088 . . . 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 17108 . . 3 ((𝑆 ∈ V ∧ 𝐸 ∈ π‘Š) β†’ dom (𝑆 sSet ⟨𝐼, 𝐸⟩) = (dom 𝑆 βˆͺ {𝐼}))
1714, 15, 16syl2anc 583 . 2 (πœ‘ β†’ dom (𝑆 sSet ⟨𝐼, 𝐸⟩) = (dom 𝑆 βˆͺ {𝐼}))
1811, 17sseqtrrd 4024 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 844   = wceq 1540   ∈ wcel 2105  Vcvv 3473   βˆͺ cun 3947   βŠ† wss 3949  {csn 4629  {cpr 4631  βŸ¨cop 4635   class class class wbr 5149  dom cdm 5677  β€˜cfv 6544  (class class class)co 7412   Struct cstr 17084   sSet csts 17101  ndxcnx 17131  Basecbs 17149
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-res 5689  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7415  df-oprab 7416  df-mpo 7417  df-struct 17085  df-sets 17102
This theorem is referenced by:  setsvtx  28559  setsiedg  28560
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