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Theorem 2strbasg 12578
Description: The base set of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.)
Hypotheses
Ref Expression
2str.g 𝐺 = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(πΈβ€˜ndx), + ⟩}
2str.e 𝐸 = Slot 𝑁
2str.l 1 < 𝑁
2str.n 𝑁 ∈ β„•
Assertion
Ref Expression
2strbasg ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š) β†’ 𝐡 = (Baseβ€˜πΊ))

Proof of Theorem 2strbasg
StepHypRef Expression
1 baseslid 12519 . 2 (Base = Slot (Baseβ€˜ndx) ∧ (Baseβ€˜ndx) ∈ β„•)
2 2str.g . . 3 𝐺 = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(πΈβ€˜ndx), + ⟩}
3 basendxnn 12518 . . . . . 6 (Baseβ€˜ndx) ∈ β„•
43a1i 9 . . . . 5 ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š) β†’ (Baseβ€˜ndx) ∈ β„•)
5 simpl 109 . . . . 5 ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š) β†’ 𝐡 ∈ 𝑉)
6 opexg 4229 . . . . 5 (((Baseβ€˜ndx) ∈ β„• ∧ 𝐡 ∈ 𝑉) β†’ ⟨(Baseβ€˜ndx), 𝐡⟩ ∈ V)
74, 5, 6syl2anc 411 . . . 4 ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š) β†’ ⟨(Baseβ€˜ndx), 𝐡⟩ ∈ V)
8 2str.e . . . . . . . 8 𝐸 = Slot 𝑁
9 2str.n . . . . . . . 8 𝑁 ∈ β„•
108, 9ndxarg 12485 . . . . . . 7 (πΈβ€˜ndx) = 𝑁
1110, 9eqeltri 2250 . . . . . 6 (πΈβ€˜ndx) ∈ β„•
1211a1i 9 . . . . 5 ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š) β†’ (πΈβ€˜ndx) ∈ β„•)
13 simpr 110 . . . . 5 ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š) β†’ + ∈ π‘Š)
14 opexg 4229 . . . . 5 (((πΈβ€˜ndx) ∈ β„• ∧ + ∈ π‘Š) β†’ ⟨(πΈβ€˜ndx), + ⟩ ∈ V)
1512, 13, 14syl2anc 411 . . . 4 ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š) β†’ ⟨(πΈβ€˜ndx), + ⟩ ∈ V)
16 prexg 4212 . . . 4 ((⟨(Baseβ€˜ndx), 𝐡⟩ ∈ V ∧ ⟨(πΈβ€˜ndx), + ⟩ ∈ V) β†’ {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(πΈβ€˜ndx), + ⟩} ∈ V)
177, 15, 16syl2anc 411 . . 3 ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š) β†’ {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(πΈβ€˜ndx), + ⟩} ∈ V)
182, 17eqeltrid 2264 . 2 ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š) β†’ 𝐺 ∈ V)
193nnrei 8928 . . . . . 6 (Baseβ€˜ndx) ∈ ℝ
20 2str.l . . . . . . 7 1 < 𝑁
21 basendx 12517 . . . . . . 7 (Baseβ€˜ndx) = 1
2220, 21, 103brtr4i 4034 . . . . . 6 (Baseβ€˜ndx) < (πΈβ€˜ndx)
2319, 22ltneii 8054 . . . . 5 (Baseβ€˜ndx) β‰  (πΈβ€˜ndx)
2423a1i 9 . . . 4 ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š) β†’ (Baseβ€˜ndx) β‰  (πΈβ€˜ndx))
25 funprg 5267 . . . 4 ((((Baseβ€˜ndx) ∈ β„• ∧ (πΈβ€˜ndx) ∈ β„•) ∧ (𝐡 ∈ 𝑉 ∧ + ∈ π‘Š) ∧ (Baseβ€˜ndx) β‰  (πΈβ€˜ndx)) β†’ Fun {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(πΈβ€˜ndx), + ⟩})
264, 12, 5, 13, 24, 25syl221anc 1249 . . 3 ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š) β†’ Fun {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(πΈβ€˜ndx), + ⟩})
272funeqi 5238 . . 3 (Fun 𝐺 ↔ Fun {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(πΈβ€˜ndx), + ⟩})
2826, 27sylibr 134 . 2 ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š) β†’ Fun 𝐺)
29 prid1g 3697 . . . 4 (⟨(Baseβ€˜ndx), 𝐡⟩ ∈ V β†’ ⟨(Baseβ€˜ndx), 𝐡⟩ ∈ {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(πΈβ€˜ndx), + ⟩})
307, 29syl 14 . . 3 ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š) β†’ ⟨(Baseβ€˜ndx), 𝐡⟩ ∈ {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(πΈβ€˜ndx), + ⟩})
3130, 2eleqtrrdi 2271 . 2 ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š) β†’ ⟨(Baseβ€˜ndx), 𝐡⟩ ∈ 𝐺)
321, 18, 28, 31strslfvd 12504 1 ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š) β†’ 𝐡 = (Baseβ€˜πΊ))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148   β‰  wne 2347  Vcvv 2738  {cpr 3594  βŸ¨cop 3596   class class class wbr 4004  Fun wfun 5211  β€˜cfv 5217  1c1 7812   < clt 7992  β„•cn 8919  ndxcnx 12459  Slot cslot 12461  Basecbs 12462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1re 7905  ax-addrcl 7908  ax-pre-ltirr 7923
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-iota 5179  df-fun 5219  df-fv 5225  df-pnf 7994  df-mnf 7995  df-ltxr 7997  df-inn 8920  df-ndx 12465  df-slot 12466  df-base 12468
This theorem is referenced by:  grpbaseg  12585  eltpsg  13543
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