MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  basprssdmsets Structured version   Visualization version   GIF version

Theorem basprssdmsets 17101
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 872 . . . 4 (πœ‘ β†’ ((Baseβ€˜ndx) ∈ dom 𝑆 ∨ (Baseβ€˜ndx) ∈ {𝐼}))
3 elun 4109 . . . 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 4621 . . . . . 6 (𝐼 ∈ π‘ˆ β†’ 𝐼 ∈ {𝐼})
75, 6syl 17 . . . . 5 (πœ‘ β†’ 𝐼 ∈ {𝐼})
87olcd 873 . . . 4 (πœ‘ β†’ (𝐼 ∈ dom 𝑆 ∨ 𝐼 ∈ {𝐼}))
9 elun 4109 . . . 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 4783 . 2 (πœ‘ β†’ {(Baseβ€˜ndx), 𝐼} βŠ† (dom 𝑆 βˆͺ {𝐼}))
12 basprssdmsets.s . . . 4 (πœ‘ β†’ 𝑆 Struct 𝑋)
13 structex 17027 . . . 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 17047 . . 3 ((𝑆 ∈ V ∧ 𝐸 ∈ π‘Š) β†’ dom (𝑆 sSet ⟨𝐼, 𝐸⟩) = (dom 𝑆 βˆͺ {𝐼}))
1714, 15, 16syl2anc 585 . 2 (πœ‘ β†’ dom (𝑆 sSet ⟨𝐼, 𝐸⟩) = (dom 𝑆 βˆͺ {𝐼}))
1811, 17sseqtrrd 3986 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 846   = wceq 1542   ∈ wcel 2107  Vcvv 3444   βˆͺ cun 3909   βŠ† wss 3911  {csn 4587  {cpr 4589  βŸ¨cop 4593   class class class wbr 5106  dom cdm 5634  β€˜cfv 6497  (class class class)co 7358   Struct cstr 17023   sSet csts 17040  ndxcnx 17070  Basecbs 17088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-res 5646  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-struct 17024  df-sets 17041
This theorem is referenced by:  setsvtx  28028  setsiedg  28029
  Copyright terms: Public domain W3C validator