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Theorem ressval3d 15764
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 3668 . . . 4 (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))
3 dfpss3 3655 . . . . 5 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐵𝐴))
43orbi1i 541 . . . 4 ((𝐴𝐵𝐴 = 𝐵) ↔ ((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∨ 𝐴 = 𝐵))
52, 4bitri 263 . . 3 (𝐴𝐵 ↔ ((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∨ 𝐴 = 𝐵))
6 simplr 788 . . . . . . 7 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → ¬ 𝐵𝐴)
7 ressval3d.s . . . . . . . 8 (𝜑𝑆𝑉)
87adantl 481 . . . . . . 7 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → 𝑆𝑉)
9 simpl 472 . . . . . . . 8 ((𝐴𝐵 ∧ ¬ 𝐵𝐴) → 𝐴𝐵)
10 ressval3d.b . . . . . . . . . 10 𝐵 = (Base‘𝑆)
11 fvex 6113 . . . . . . . . . 10 (Base‘𝑆) ∈ V
1210, 11eqeltri 2684 . . . . . . . . 9 𝐵 ∈ V
1312a1i 11 . . . . . . . 8 (𝜑𝐵 ∈ V)
14 ssexg 4732 . . . . . . . 8 ((𝐴𝐵𝐵 ∈ V) → 𝐴 ∈ V)
159, 13, 14syl2an 493 . . . . . . 7 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → 𝐴 ∈ V)
16 ressval3d.r . . . . . . . 8 𝑅 = (𝑆s 𝐴)
1716, 10ressval2 15756 . . . . . . 7 ((¬ 𝐵𝐴𝑆𝑉𝐴 ∈ V) → 𝑅 = (𝑆 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩))
186, 8, 15, 17syl3anc 1318 . . . . . 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 3554 . . . . . . . . . . . . 13 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
2221biimpi 205 . . . . . . . . . . . 12 (𝐴𝐵 → (𝐴𝐵) = 𝐴)
2322eqcomd 2616 . . . . . . . . . . 11 (𝐴𝐵𝐴 = (𝐴𝐵))
2423adantr 480 . . . . . . . . . 10 ((𝐴𝐵 ∧ ¬ 𝐵𝐴) → 𝐴 = (𝐴𝐵))
2524adantr 480 . . . . . . . . 9 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → 𝐴 = (𝐴𝐵))
2620, 25opeq12d 4348 . . . . . . . 8 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → ⟨𝐸, 𝐴⟩ = ⟨(Base‘ndx), (𝐴𝐵)⟩)
2726eqcomd 2616 . . . . . . 7 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → ⟨(Base‘ndx), (𝐴𝐵)⟩ = ⟨𝐸, 𝐴⟩)
2827oveq2d 6565 . . . . . 6 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → (𝑆 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩) = (𝑆 sSet ⟨𝐸, 𝐴⟩))
2918, 28eqtrd 2644 . . . . 5 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∧ 𝜑) → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
3029ex 449 . . . 4 ((𝐴𝐵 ∧ ¬ 𝐵𝐴) → (𝜑𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
3116a1i 11 . . . . . . 7 ((𝐴 = 𝐵𝜑) → 𝑅 = (𝑆s 𝐴))
32 oveq2 6557 . . . . . . . 8 (𝐴 = 𝐵 → (𝑆s 𝐴) = (𝑆s 𝐵))
3332adantr 480 . . . . . . 7 ((𝐴 = 𝐵𝜑) → (𝑆s 𝐴) = (𝑆s 𝐵))
347adantl 481 . . . . . . . 8 ((𝐴 = 𝐵𝜑) → 𝑆𝑉)
3510ressid 15762 . . . . . . . 8 (𝑆𝑉 → (𝑆s 𝐵) = 𝑆)
3634, 35syl 17 . . . . . . 7 ((𝐴 = 𝐵𝜑) → (𝑆s 𝐵) = 𝑆)
3731, 33, 363eqtrd 2648 . . . . . 6 ((𝐴 = 𝐵𝜑) → 𝑅 = 𝑆)
38 df-base 15700 . . . . . . . 8 Base = Slot 1
39 1nn 10908 . . . . . . . 8 1 ∈ ℕ
40 ressval3d.f . . . . . . . 8 (𝜑 → Fun 𝑆)
41 ressval3d.d . . . . . . . . 9 (𝜑𝐸 ∈ dom 𝑆)
4219, 41syl5eqelr 2693 . . . . . . . 8 (𝜑 → (Base‘ndx) ∈ dom 𝑆)
4338, 39, 7, 40, 42setsidvald 15721 . . . . . . 7 (𝜑𝑆 = (𝑆 sSet ⟨(Base‘ndx), (Base‘𝑆)⟩))
4443adantl 481 . . . . . 6 ((𝐴 = 𝐵𝜑) → 𝑆 = (𝑆 sSet ⟨(Base‘ndx), (Base‘𝑆)⟩))
4519a1i 11 . . . . . . . . 9 ((𝐴 = 𝐵𝜑) → 𝐸 = (Base‘ndx))
46 simpl 472 . . . . . . . . . 10 ((𝐴 = 𝐵𝜑) → 𝐴 = 𝐵)
4746, 10syl6eq 2660 . . . . . . . . 9 ((𝐴 = 𝐵𝜑) → 𝐴 = (Base‘𝑆))
4845, 47opeq12d 4348 . . . . . . . 8 ((𝐴 = 𝐵𝜑) → ⟨𝐸, 𝐴⟩ = ⟨(Base‘ndx), (Base‘𝑆)⟩)
4948eqcomd 2616 . . . . . . 7 ((𝐴 = 𝐵𝜑) → ⟨(Base‘ndx), (Base‘𝑆)⟩ = ⟨𝐸, 𝐴⟩)
5049oveq2d 6565 . . . . . 6 ((𝐴 = 𝐵𝜑) → (𝑆 sSet ⟨(Base‘ndx), (Base‘𝑆)⟩) = (𝑆 sSet ⟨𝐸, 𝐴⟩))
5137, 44, 503eqtrd 2648 . . . . 5 ((𝐴 = 𝐵𝜑) → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
5251ex 449 . . . 4 (𝐴 = 𝐵 → (𝜑𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
5330, 52jaoi 393 . . 3 (((𝐴𝐵 ∧ ¬ 𝐵𝐴) ∨ 𝐴 = 𝐵) → (𝜑𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
545, 53sylbi 206 . 2 (𝐴𝐵 → (𝜑𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩)))
551, 54mpcom 37 1 (𝜑𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wa 383   = wceq 1475  wcel 1977  Vcvv 3173  cin 3539  wss 3540  wpss 3541  cop 4131  dom cdm 5038  Fun wfun 5798  cfv 5804  (class class class)co 6549  1c1 9816  ndxcnx 15692   sSet csts 15693  Basecbs 15695  s cress 15696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-i2m1 9883  ax-1ne0 9884  ax-rrecex 9887  ax-cnre 9888
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-nn 10898  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702
This theorem is referenced by:  estrres  16602
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