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Theorem ressval3d 15858
Description: Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.)
Hypotheses
Ref Expression
ressval3d.r 𝑅 = (𝑆s 𝐴)
ressval3d.b 𝐵 = (Base‘𝑆)
ressval3d.e 𝐸 = (Base‘ndx)
ressval3d.s (𝜑𝑆𝑉)
ressval3d.f (𝜑 → Fun 𝑆)
ressval3d.d (𝜑𝐸 ∈ dom 𝑆)
ressval3d.a (𝜑𝐴𝑌)
ressval3d.u (𝜑𝐴𝐵)
Assertion
Ref Expression
ressval3d (𝜑𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))

Proof of Theorem ressval3d
StepHypRef Expression
1 ressval3d.u . 2 (𝜑𝐴𝐵)
2 sspss 3684 . . . 4 (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))
3 dfpss3 3671 . . . . 5 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐵𝐴))
43orbi1i 542 . . . 4 ((𝐴𝐵𝐴 = 𝐵) ↔ ((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∨ 𝐴 = 𝐵))
52, 4bitri 264 . . 3 (𝐴𝐵 ↔ ((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∨ 𝐴 = 𝐵))
6 simplr 791 . . . . . . 7 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → ¬ 𝐵𝐴)
7 ressval3d.s . . . . . . . 8 (𝜑𝑆𝑉)
87adantl 482 . . . . . . 7 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → 𝑆𝑉)
9 simpl 473 . . . . . . . 8 ((𝐴𝐵 ∧ ¬ 𝐵𝐴) → 𝐴𝐵)
10 ressval3d.b . . . . . . . . . 10 𝐵 = (Base‘𝑆)
11 fvex 6158 . . . . . . . . . 10 (Base‘𝑆) ∈ V
1210, 11eqeltri 2694 . . . . . . . . 9 𝐵 ∈ V
1312a1i 11 . . . . . . . 8 (𝜑𝐵 ∈ V)
14 ssexg 4764 . . . . . . . 8 ((𝐴𝐵𝐵 ∈ V) → 𝐴 ∈ V)
159, 13, 14syl2an 494 . . . . . . 7 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → 𝐴 ∈ V)
16 ressval3d.r . . . . . . . 8 𝑅 = (𝑆s 𝐴)
1716, 10ressval2 15850 . . . . . . 7 ((¬ 𝐵𝐴𝑆𝑉𝐴 ∈ V) → 𝑅 = (𝑆 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩))
186, 8, 15, 17syl3anc 1323 . . . . . 6 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → 𝑅 = (𝑆 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩))
19 ressval3d.e . . . . . . . . . 10 𝐸 = (Base‘ndx)
2019a1i 11 . . . . . . . . 9 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → 𝐸 = (Base‘ndx))
21 df-ss 3569 . . . . . . . . . . . . 13 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
2221biimpi 206 . . . . . . . . . . . 12 (𝐴𝐵 → (𝐴𝐵) = 𝐴)
2322eqcomd 2627 . . . . . . . . . . 11 (𝐴𝐵𝐴 = (𝐴𝐵))
2423adantr 481 . . . . . . . . . 10 ((𝐴𝐵 ∧ ¬ 𝐵𝐴) → 𝐴 = (𝐴𝐵))
2524adantr 481 . . . . . . . . 9 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → 𝐴 = (𝐴𝐵))
2620, 25opeq12d 4378 . . . . . . . 8 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → ⟨𝐸, 𝐴⟩ = ⟨(Base‘ndx), (𝐴𝐵)⟩)
2726eqcomd 2627 . . . . . . 7 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → ⟨(Base‘ndx), (𝐴𝐵)⟩ = ⟨𝐸, 𝐴⟩)
2827oveq2d 6620 . . . . . 6 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → (𝑆 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩) = (𝑆 sSet ⟨𝐸, 𝐴⟩))
2918, 28eqtrd 2655 . . . . 5 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
3029ex 450 . . . 4 ((𝐴𝐵 ∧ ¬ 𝐵𝐴) → (𝜑𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
3116a1i 11 . . . . . . 7 ((𝐴 = 𝐵𝜑) → 𝑅 = (𝑆s 𝐴))
32 oveq2 6612 . . . . . . . 8 (𝐴 = 𝐵 → (𝑆s 𝐴) = (𝑆s 𝐵))
3332adantr 481 . . . . . . 7 ((𝐴 = 𝐵𝜑) → (𝑆s 𝐴) = (𝑆s 𝐵))
347adantl 482 . . . . . . . 8 ((𝐴 = 𝐵𝜑) → 𝑆𝑉)
3510ressid 15856 . . . . . . . 8 (𝑆𝑉 → (𝑆s 𝐵) = 𝑆)
3634, 35syl 17 . . . . . . 7 ((𝐴 = 𝐵𝜑) → (𝑆s 𝐵) = 𝑆)
3731, 33, 363eqtrd 2659 . . . . . 6 ((𝐴 = 𝐵𝜑) → 𝑅 = 𝑆)
38 df-base 15786 . . . . . . . 8 Base = Slot 1
39 1nn 10975 . . . . . . . 8 1 ∈ ℕ
40 ressval3d.f . . . . . . . 8 (𝜑 → Fun 𝑆)
41 ressval3d.d . . . . . . . . 9 (𝜑𝐸 ∈ dom 𝑆)
4219, 41syl5eqelr 2703 . . . . . . . 8 (𝜑 → (Base‘ndx) ∈ dom 𝑆)
4338, 39, 7, 40, 42setsidvald 15810 . . . . . . 7 (𝜑𝑆 = (𝑆 sSet ⟨(Base‘ndx), (Base‘𝑆)⟩))
4443adantl 482 . . . . . 6 ((𝐴 = 𝐵𝜑) → 𝑆 = (𝑆 sSet ⟨(Base‘ndx), (Base‘𝑆)⟩))
4519a1i 11 . . . . . . . . 9 ((𝐴 = 𝐵𝜑) → 𝐸 = (Base‘ndx))
46 simpl 473 . . . . . . . . . 10 ((𝐴 = 𝐵𝜑) → 𝐴 = 𝐵)
4746, 10syl6eq 2671 . . . . . . . . 9 ((𝐴 = 𝐵𝜑) → 𝐴 = (Base‘𝑆))
4845, 47opeq12d 4378 . . . . . . . 8 ((𝐴 = 𝐵𝜑) → ⟨𝐸, 𝐴⟩ = ⟨(Base‘ndx), (Base‘𝑆)⟩)
4948eqcomd 2627 . . . . . . 7 ((𝐴 = 𝐵𝜑) → ⟨(Base‘ndx), (Base‘𝑆)⟩ = ⟨𝐸, 𝐴⟩)
5049oveq2d 6620 . . . . . 6 ((𝐴 = 𝐵𝜑) → (𝑆 sSet ⟨(Base‘ndx), (Base‘𝑆)⟩) = (𝑆 sSet ⟨𝐸, 𝐴⟩))
5137, 44, 503eqtrd 2659 . . . . 5 ((𝐴 = 𝐵𝜑) → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
5251ex 450 . . . 4 (𝐴 = 𝐵 → (𝜑𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
5330, 52jaoi 394 . . 3 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∨ 𝐴 = 𝐵) → (𝜑𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
545, 53sylbi 207 . 2 (𝐴𝐵 → (𝜑𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
551, 54mpcom 38 1 (𝜑𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 383  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  cin 3554  wss 3555  wpss 3556  cop 4154  dom cdm 5074  Fun wfun 5841  cfv 5847  (class class class)co 6604  1c1 9881  ndxcnx 15778   sSet csts 15779  Basecbs 15781  s cress 15782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-i2m1 9948  ax-1ne0 9949  ax-rrecex 9952  ax-cnre 9953
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-nn 10965  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788
This theorem is referenced by:  estrres  16700
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