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Theorem 1strwunbndx 17107
Description: A constructed one-slot structure in a weak universe containing the index of the base set extractor. (Contributed by AV, 27-Mar-2020.)
Hypotheses
Ref Expression
1str.g 𝐺 = {⟨(Baseβ€˜ndx), 𝐡⟩}
1strwun.u (πœ‘ β†’ π‘ˆ ∈ WUni)
1strwunbndx.b (πœ‘ β†’ (Baseβ€˜ndx) ∈ π‘ˆ)
Assertion
Ref Expression
1strwunbndx ((πœ‘ ∧ 𝐡 ∈ π‘ˆ) β†’ 𝐺 ∈ π‘ˆ)

Proof of Theorem 1strwunbndx
StepHypRef Expression
1 1str.g . 2 𝐺 = {⟨(Baseβ€˜ndx), 𝐡⟩}
2 1strwun.u . . . 4 (πœ‘ β†’ π‘ˆ ∈ WUni)
32adantr 482 . . 3 ((πœ‘ ∧ 𝐡 ∈ π‘ˆ) β†’ π‘ˆ ∈ WUni)
4 1strwunbndx.b . . . . 5 (πœ‘ β†’ (Baseβ€˜ndx) ∈ π‘ˆ)
54adantr 482 . . . 4 ((πœ‘ ∧ 𝐡 ∈ π‘ˆ) β†’ (Baseβ€˜ndx) ∈ π‘ˆ)
6 simpr 486 . . . 4 ((πœ‘ ∧ 𝐡 ∈ π‘ˆ) β†’ 𝐡 ∈ π‘ˆ)
73, 5, 6wunop 10663 . . 3 ((πœ‘ ∧ 𝐡 ∈ π‘ˆ) β†’ ⟨(Baseβ€˜ndx), 𝐡⟩ ∈ π‘ˆ)
83, 7wunsn 10657 . 2 ((πœ‘ ∧ 𝐡 ∈ π‘ˆ) β†’ {⟨(Baseβ€˜ndx), 𝐡⟩} ∈ π‘ˆ)
91, 8eqeltrid 2838 1 ((πœ‘ ∧ 𝐡 ∈ π‘ˆ) β†’ 𝐺 ∈ π‘ˆ)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {csn 4587  βŸ¨cop 4593  β€˜cfv 6497  WUnicwun 10641  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-ext 2704
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-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-v 3446  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-tr 5224  df-wun 10643
This theorem is referenced by:  1strwun  17108  1strwunOLD  17109  equivestrcsetc  18045
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