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Theorem basprssdmsets 16132
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 862 . . . 4 (𝜑 → ((Base‘ndx) ∈ dom 𝑆 ∨ (Base‘ndx) ∈ {𝐼}))
3 elun 3904 . . . 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 4345 . . . . . 6 (𝐼𝑈𝐼 ∈ {𝐼})
75, 6syl 17 . . . . 5 (𝜑𝐼 ∈ {𝐼})
87olcd 863 . . . 4 (𝜑 → (𝐼 ∈ dom 𝑆𝐼 ∈ {𝐼}))
9 elun 3904 . . . 4 (𝐼 ∈ (dom 𝑆 ∪ {𝐼}) ↔ (𝐼 ∈ dom 𝑆𝐼 ∈ {𝐼}))
108, 9sylibr 224 . . 3 (𝜑𝐼 ∈ (dom 𝑆 ∪ {𝐼}))
114, 10prssd 4488 . 2 (𝜑 → {(Base‘ndx), 𝐼} ⊆ (dom 𝑆 ∪ {𝐼}))
12 basprssdmsets.s . . . 4 (𝜑𝑆 Struct 𝑋)
13 structex 16075 . . . 4 (𝑆 Struct 𝑋𝑆 ∈ V)
1412, 13syl 17 . . 3 (𝜑𝑆 ∈ V)
15 basprssdmsets.w . . 3 (𝜑𝐸𝑊)
16 setsdm 16099 . . 3 ((𝑆 ∈ V ∧ 𝐸𝑊) → dom (𝑆 sSet ⟨𝐼, 𝐸⟩) = (dom 𝑆 ∪ {𝐼}))
1714, 15, 16syl2anc 573 . 2 (𝜑 → dom (𝑆 sSet ⟨𝐼, 𝐸⟩) = (dom 𝑆 ∪ {𝐼}))
1811, 17sseqtr4d 3791 1 (𝜑 → {(Base‘ndx), 𝐼} ⊆ dom (𝑆 sSet ⟨𝐼, 𝐸⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 836   = wceq 1631  wcel 2145  Vcvv 3351  cun 3721  wss 3723  {csn 4316  {cpr 4318  cop 4322   class class class wbr 4786  dom cdm 5249  cfv 6031  (class class class)co 6793   Struct cstr 16060  ndxcnx 16061   sSet csts 16062  Basecbs 16064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-res 5261  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-struct 16066  df-sets 16071
This theorem is referenced by:  setsvtx  26148  setsiedg  26149
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